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

<b>let </b><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:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">r</font> being  <a href="supinf_1.html#NM3" title="SUPINF_1:NM.3">R_eal</a>  st <font color="Olive" title="b1">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  &amp; ( for <font color="Olive" title="b2">n</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">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="b2">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">r</font> <a href="supinf_2.html#K2" title="SUPINF_2:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">seq2</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Olive" title="b2">n</font></span>)</span> ) holds <br/>(  <a href="rinfsup2.html#K6" title="RINFSUP2:func.6">lim_inf</a> <font color="Maroon" title="c1">seq1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">r</font> <a href="supinf_2.html#K2" title="SUPINF_2:func.2">+</a> <span class="p1">(<span class="default"><a href="rinfsup2.html#K6" title="RINFSUP2:func.6">lim_inf</a> <font color="Maroon" title="c2">seq2</font></span>)</span> &amp;  <a href="rinfsup2.html#K5" title="RINFSUP2:func.5">lim_sup</a> <font color="Maroon" title="c1">seq1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">r</font> <a href="supinf_2.html#K2" title="SUPINF_2:func.2">+</a> <span class="p1">(<span class="default"><a href="rinfsup2.html#K5" title="RINFSUP2:func.5">lim_sup</a> <font color="Maroon" title="c2">seq2</font></span>)</span> )</span><br/><b>let </b><font color="Maroon" title="c3">r</font> be   <a href="supinf_1.html#NM3" title="SUPINF_1:NM.3">R_eal</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c3">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  &amp; ( for <font color="Olive" title="b1">n</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">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="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">r</font> <a href="supinf_2.html#K2" title="SUPINF_2:func.2">+</a> <span class="p1">(<span class="default"><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></span>)</span> ) implies (  <a href="rinfsup2.html#K6" title="RINFSUP2:func.6">lim_inf</a> <font color="Maroon" title="c1">seq1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">r</font> <a href="supinf_2.html#K2" title="SUPINF_2:func.2">+</a> <span class="p1">(<span class="default"><a href="rinfsup2.html#K6" title="RINFSUP2:func.6">lim_inf</a> <font color="Maroon" title="c2">seq2</font></span>)</span> &amp;  <a href="rinfsup2.html#K5" title="RINFSUP2:func.5">lim_sup</a> <font color="Maroon" title="c1">seq1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">r</font> <a href="supinf_2.html#K2" title="SUPINF_2:func.2">+</a> <span class="p1">(<span class="default"><a href="rinfsup2.html#K5" title="RINFSUP2:func.5">lim_sup</a> <font color="Maroon" title="c2">seq2</font></span>)</span> ) )</span><br/>





<b>assume </b><b>that </b><br/><a NAME="E1:10"/><i><font color="Green" title="E11">A0</font></i>: 
<font color="Maroon" title="c3">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> 
 <b>and </b><br/><a NAME="E2:10"/><i><font color="Green" title="E12">A1</font></i>: 
for <font color="Olive" title="b1">n</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">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="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">r</font> <a href="supinf_2.html#K2" title="SUPINF_2:func.2">+</a> <span class="p1">(<span class="default"><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></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  <a href="rinfsup2.html#K6" title="RINFSUP2:func.6">lim_inf</a> <font color="Maroon" title="c1">seq1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">r</font> <a href="supinf_2.html#K2" title="SUPINF_2:func.2">+</a> <span class="p1">(<span class="default"><a href="rinfsup2.html#K6" title="RINFSUP2:func.6">lim_inf</a> <font color="Maroon" title="c2">seq2</font></span>)</span> &amp;  <a href="rinfsup2.html#K5" title="RINFSUP2:func.5">lim_sup</a> <font color="Maroon" title="c1">seq1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">r</font> <a href="supinf_2.html#K2" title="SUPINF_2:func.2">+</a> <span class="p1">(<span class="default"><a href="rinfsup2.html#K5" title="RINFSUP2:func.5">lim_sup</a> <font color="Maroon" title="c2">seq2</font></span>)</span> )</span><br/>

<b>reconsider </b><font color="Maroon" title="c4">R1</font> =  <a href="supinf_2.html#K18" title="SUPINF_2:func.18">rng</a> <span class="p1">(<span class="default"><a href="rinfsup2.html#K3" title="RINFSUP2:func.3">inferior_realsequence</a> <font color="Maroon" title="c1">seq1</font></span>)</span>, <font color="Maroon" title="c5">R2</font> =  <a href="supinf_2.html#K18" title="SUPINF_2:func.18">rng</a> <span class="p1">(<span class="default"><a href="rinfsup2.html#K3" title="RINFSUP2:func.3">inferior_realsequence</a> <font color="Maroon" title="c2">seq2</font></span>)</span>, <font color="Maroon" title="c6">P1</font> =  <a href="supinf_2.html#K18" title="SUPINF_2:func.18">rng</a> <span class="p1">(<span class="default"><a href="rinfsup2.html#K4" title="RINFSUP2:func.4">superior_realsequence</a> <font color="Maroon" title="c1">seq1</font></span>)</span>, <font color="Maroon" title="c7">P2</font> =  <a href="supinf_2.html#K18" title="SUPINF_2:func.18">rng</a> <span class="p1">(<span class="default"><a href="rinfsup2.html#K4" title="RINFSUP2:func.4">superior_realsequence</a> <font color="Maroon" title="c2">seq2</font></span>)</span> as  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <a href="numbers.html#K7" title="NUMBERS:func.7">ExtREAL</a>  ;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="10_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c8">z</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c8">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">R1</font> implies <font color="Maroon" title="c8">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c3">r</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="supinf_2.html#K5" title="SUPINF_2:func.5">+</a> <font color="Maroon" title="c5">R2</font> )</span><br/><b>assume </b><a NAME="E1:10_1"/>
<font color="Maroon" title="c8">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">R1</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c8">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c3">r</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="supinf_2.html#K5" title="SUPINF_2:func.5">+</a> <font color="Maroon" title="c5">R2</font></span><br/><b>then consider </b><font color="Maroon" title="c9">n</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:10_1"/><i><font color="Green" title="E13">B1</font></i>: 
( <font color="Maroon" title="c9">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  &amp; <span class="p1">(<span class="default"><a href="rinfsup2.html#K3" title="RINFSUP2:func.3">inferior_realsequence</a> <font color="Maroon" title="c1">seq1</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">z</font> )
 <b>by </b><i><a class="ref" href="funct_2.html#T17" title="FUNCT_2:th.17">FUNCT_2:17</a></i>;<br/><b>reconsider </b><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>  <b>by </b><i><a class="txt" href="#E3:10_1"><i><font color="Green" title="E13">B1</font></i></a></i>;<br/><b>consider </b><font color="Maroon" title="c11">Y1</font> being  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <a href="numbers.html#K7" title="NUMBERS:func.7">ExtREAL</a> <b> such that </b><br/><a NAME="E6:10_1"/><i><font color="Green" title="E14">B2</font></i>: 
( <font color="Maroon" title="c11">Y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon" title="c1">seq1</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Olive" title="b1">k</font></span>)</span> where <font color="Olive" title="b1">k</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>  : <font color="Maroon" title="c10">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">k</font> </span>}</span>  &amp; <font color="Maroon" title="c8">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_2.html#K7" title="SUPINF_2:func.7">inf</a> <font color="Maroon" title="c11">Y1</font> )
 <b>by </b><i><a class="txt" href="#E3:10_1"><i><font color="Green" title="E13">B1</font></i></a>, <a class="ref" href="rinfsup2.html#D6" title="RINFSUP2:def.6">RINFSUP2:def 6</a></i>;<br/><b>consider </b><font color="Maroon" title="c12">Y2</font> being  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <a href="numbers.html#K7" title="NUMBERS:func.7">ExtREAL</a> <b> such that </b><br/><a NAME="E8:10_1"/><i><font color="Green" title="E15">B21</font></i>: 
( <font color="Maroon" title="c12">Y2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon" title="c2">seq2</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Olive" title="b1">k</font></span>)</span> where <font color="Olive" title="b1">k</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>  : <font color="Maroon" title="c10">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">k</font> </span>}</span>  &amp; <span class="p1">(<span class="default"><a href="rinfsup2.html#K3" title="RINFSUP2:func.3">inferior_realsequence</a> <font color="Maroon" title="c2">seq2</font></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c10">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_2.html#K7" title="SUPINF_2:func.7">inf</a> <font color="Maroon" title="c12">Y2</font> )
 <b>by </b><i><a class="ref" href="rinfsup2.html#D6" title="RINFSUP2:def.6">RINFSUP2:def 6</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="10_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c13">w</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c13">w</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">Y1</font> implies <font color="Maroon" title="c13">w</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c3">r</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="supinf_2.html#K5" title="SUPINF_2:func.5">+</a> <font color="Maroon" title="c12">Y2</font> )</span><br/><b>assume </b><a NAME="E1:10_1_1"/>
<font color="Maroon" title="c13">w</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">Y1</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c13">w</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c3">r</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="supinf_2.html#K5" title="SUPINF_2:func.5">+</a> <font color="Maroon" title="c12">Y2</font></span><br/><b>then consider </b><font color="Maroon" title="c14">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> <b> such that </b><br/><a NAME="E3:10_1_1"/><i><font color="Green" title="E16">B3</font></i>: 
( <font color="Maroon" title="c13">w</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> <font color="Maroon" title="c14">k1</font> &amp; <font color="Maroon" title="c10">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c14">k1</font> )
 <b>by </b><i><a class="txt" href="#E6:10_1"><i><font color="Green" title="E14">B2</font></i></a></i>;<br/><b>reconsider </b><font color="Maroon" title="c15">w1</font> = <font color="Maroon" title="c13">w</font> as   <a href="supinf_1.html#NM3" title="SUPINF_1:NM.3">R_eal</a> <b>by </b><i><a class="txt" href="#E3:10_1_1"><i><font color="Green" title="E16">B3</font></i></a></i>;<br/><a NAME="E5:10_1_1"/><i><font color="Green" title="E17">B4</font></i>: 
<font color="Maroon" title="c15">w1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">r</font> <a href="supinf_2.html#K2" title="SUPINF_2:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">seq2</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c14">k1</font></span>)</span>
 <b>by </b><i><a class="txt" href="#E2:10"><i><font color="Green" title="E12">A1</font></i></a>, <a class="txt" href="#E3:10_1_1"><i><font color="Green" title="E16">B3</font></i></a></i>;<br/><a NAME="E6:10_1_1"/>
( <font color="Maroon" title="c3">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c3">r</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> &amp; <font color="Maroon" title="c2">seq2</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c14">k1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c12">Y2</font> )
 <b>by </b><i><a class="txt" href="#E3:10_1_1"><i><font color="Green" title="E16">B3</font></i></a>, <a class="txt" href="#E8:10_1"><i><font color="Green" title="E15">B21</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/><b>hence </b><a NAME="E7:10_1_1"/>
<font color="Maroon" title="c13">w</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c3">r</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="supinf_2.html#K5" title="SUPINF_2:func.5">+</a> <font color="Maroon" title="c12">Y2</font>
 <b>by </b><i><a class="txt" href="#E5:10_1_1"><i><font color="Green" title="E17">B4</font></i></a>, <a class="ref" href="supinf_2.html#D5" title="SUPINF_2:def.5">SUPINF_2:def 5</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><a NAME="E10:10_1"/><b>then </b><i><font color="Green" title="E16">B6</font></i>: 
<font color="Maroon" title="c11">Y1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c3">r</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="supinf_2.html#K5" title="SUPINF_2:func.5">+</a> <font color="Maroon" title="c12">Y2</font>
 <b>by </b><i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="10_1_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c13">w</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c13">w</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c3">r</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="supinf_2.html#K5" title="SUPINF_2:func.5">+</a> <font color="Maroon" title="c12">Y2</font> implies <font color="Maroon" title="c13">w</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">Y1</font> )</span><br/><b>assume </b><a NAME="E1:10_1_2"/>
<font color="Maroon" title="c13">w</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c3">r</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="supinf_2.html#K5" title="SUPINF_2:func.5">+</a> <font color="Maroon" title="c12">Y2</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c13">w</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">Y1</font></span><br/><b>then consider </b><font color="Maroon" title="c14">w1</font>, <font color="Maroon" title="c15">w2</font> being   <a href="supinf_1.html#NM3" title="SUPINF_1:NM.3">R_eal</a><b> such that </b><br/><a NAME="E3:10_1_2"/><i><font color="Green" title="E17">B7</font></i>: 
( <font color="Maroon" title="c14">w1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c3">r</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> &amp; <font color="Maroon" title="c15">w2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c12">Y2</font> &amp; <font color="Maroon" title="c13">w</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">w1</font> <a href="supinf_2.html#K2" title="SUPINF_2:func.2">+</a> <font color="Maroon" title="c15">w2</font> )
 <b>by </b><i><a class="ref" href="supinf_2.html#D5" title="SUPINF_2:def.5">SUPINF_2:def 5</a></i>;<br/><b>consider </b><font color="Maroon" title="c16">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> <b> such that </b><br/><a NAME="E5:10_1_2"/><i><font color="Green" title="E18">B8</font></i>: 
( <font color="Maroon" title="c15">w2</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> <font color="Maroon" title="c16">k2</font> &amp; <font color="Maroon" title="c10">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c16">k2</font> )
 <b>by </b><i><a class="txt" href="#E8:10_1"><i><font color="Green" title="E15">B21</font></i></a>, <a class="txt" href="#E3:10_1_2"><i><font color="Green" title="E17">B7</font></i></a></i>;<br/><a NAME="E6:10_1_2"/>
<font color="Maroon" title="c14">w1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">r</font>
 <b>by </b><i><a class="txt" href="#E3:10_1_2"><i><font color="Green" title="E17">B7</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/><a NAME="E7:10_1_2"/><b>then </b>
<font color="Maroon" title="c13">w</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> <font color="Maroon" title="c16">k2</font>
 <b>by </b><i><a class="txt" href="#E2:10"><i><font color="Green" title="E12">A1</font></i></a>, <a class="txt" href="#E3:10_1_2"><i><font color="Green" title="E17">B7</font></i></a>, <a class="txt" href="#E5:10_1_2"><i><font color="Green" title="E18">B8</font></i></a></i>;<br/><b>hence </b><a NAME="E8:10_1_2"/>
<font color="Maroon" title="c13">w</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">Y1</font>
 <b>by </b><i><a class="txt" href="#E6:10_1"><i><font color="Green" title="E14">B2</font></i></a>, <a class="txt" href="#E5:10_1_2"><i><font color="Green" title="E18">B8</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><a NAME="E12:10_1"/><b>then </b>
<span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c3">r</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="supinf_2.html#K5" title="SUPINF_2:func.5">+</a> <font color="Maroon" title="c12">Y2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c11">Y1</font>
 <b>by </b><i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></i>;<br/><a NAME="E13:10_1"/><b>then </b>
<font color="Maroon" title="c11">Y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c3">r</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="supinf_2.html#K5" title="SUPINF_2:func.5">+</a> <font color="Maroon" title="c12">Y2</font>
 <b>by </b><i><a class="txt" href="#E10:10_1"><i><font color="Green" title="E16">B6</font></i></a>, <a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a></i>;<br/><a NAME="E14:10_1"/><b>then </b>
 <a href="supinf_2.html#K7" title="SUPINF_2:func.7">inf</a> <font color="Maroon" title="c11">Y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K7" title="SUPINF_2:func.7">inf</a> <span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c3">r</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span> <a href="supinf_2.html#K2" title="SUPINF_2:func.2">+</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K7" title="SUPINF_2:func.7">inf</a> <font color="Maroon" title="c12">Y2</font></span>)</span>
 <b>by </b><i><a class="txt" href="#E1:10"><i><font color="Green" title="E11">A0</font></i></a>, <a class="ref" href="mesfun10.html#T9" target="_self" title="MESFUN10:th.9">SUPINF227a</a></i>;<br/><a NAME="E15:10_1"/><b>then </b><i><font color="Green" title="E17">B9</font></i>: 
 <a href="supinf_2.html#K7" title="SUPINF_2:func.7">inf</a> <font color="Maroon" title="c11">Y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">r</font> <a href="supinf_2.html#K2" title="SUPINF_2:func.2">+</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K7" title="SUPINF_2:func.7">inf</a> <font color="Maroon" title="c12">Y2</font></span>)</span>
 <b>by </b><i><a class="ref" href="xxreal_2.html#T13" title="XXREAL_2:th.13">XXREAL_2:13</a></i>;<br/><a NAME="E16:10_1"/>
( <font color="Maroon" title="c3">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c3">r</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> &amp; <span class="p1">(<span class="default"><a href="rinfsup2.html#K3" title="RINFSUP2:func.3">inferior_realsequence</a> <font color="Maroon" title="c2">seq2</font></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c10">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">R2</font> )
 <b>by </b><i><a class="ref" href="funct_2.html#T6" title="FUNCT_2:th.6">FUNCT_2:6</a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/><b>hence </b><a NAME="E17:10_1"/>
<font color="Maroon" title="c8">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c3">r</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="supinf_2.html#K5" title="SUPINF_2:func.5">+</a> <font color="Maroon" title="c5">R2</font>
 <b>by </b><i><a class="txt" href="#E15:10_1"><i><font color="Green" title="E17">B9</font></i></a>, <a class="txt" href="#E6:10_1"><i><font color="Green" title="E14">B2</font></i></a>, <a class="txt" href="#E8:10_1"><i><font color="Green" title="E15">B21</font></i></a>, <a class="ref" href="supinf_2.html#D5" title="SUPINF_2:def.5">SUPINF_2:def 5</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div>
<a NAME="E5:10"/><b>then </b><i><font color="Green" title="E13">B9</font></i>: 
<font color="Maroon" title="c4">R1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c3">r</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="supinf_2.html#K5" title="SUPINF_2:func.5">+</a> <font color="Maroon" title="c5">R2</font>
 
<b>by </b><i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="10_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c8">z</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c8">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c3">r</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="supinf_2.html#K5" title="SUPINF_2:func.5">+</a> <font color="Maroon" title="c5">R2</font> implies <font color="Maroon" title="c8">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">R1</font> )</span><br/><b>assume </b><a NAME="E1:10_2"/>
<font color="Maroon" title="c8">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c3">r</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="supinf_2.html#K5" title="SUPINF_2:func.5">+</a> <font color="Maroon" title="c5">R2</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c8">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">R1</font></span><br/><b>then consider </b><font color="Maroon" title="c9">z2</font>, <font color="Maroon" title="c10">z3</font> being   <a href="supinf_1.html#NM3" title="SUPINF_1:NM.3">R_eal</a><b> such that </b><br/><a NAME="E3:10_2"/><i><font color="Green" title="E14">C2</font></i>: 
( <font color="Maroon" title="c9">z2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c3">r</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> &amp; <font color="Maroon" title="c10">z3</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">R2</font> &amp; <font color="Maroon" title="c8">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">z2</font> <a href="supinf_2.html#K2" title="SUPINF_2:func.2">+</a> <font color="Maroon" title="c10">z3</font> )
 <b>by </b><i><a class="ref" href="supinf_2.html#D5" title="SUPINF_2:def.5">SUPINF_2:def 5</a></i>;<br/><b>consider </b><font color="Maroon" title="c11">n</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E5:10_2"/><i><font color="Green" title="E15">C3</font></i>: 
( <font color="Maroon" title="c11">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  &amp; <span class="p1">(<span class="default"><a href="rinfsup2.html#K3" title="RINFSUP2:func.3">inferior_realsequence</a> <font color="Maroon" title="c2">seq2</font></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c11">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">z3</font> )
 <b>by </b><i><a class="txt" href="#E3:10_2"><i><font color="Green" title="E14">C2</font></i></a>, <a class="ref" href="funct_2.html#T17" title="FUNCT_2:th.17">FUNCT_2:17</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c12">n</font> = <font color="Maroon" title="c11">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>  <b>by </b><i><a class="txt" href="#E5:10_2"><i><font color="Green" title="E15">C3</font></i></a></i>;<br/><b>consider </b><font color="Maroon" title="c13">Y2</font> being  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <a href="numbers.html#K7" title="NUMBERS:func.7">ExtREAL</a> <b> such that </b><br/><a NAME="E8:10_2"/><i><font color="Green" title="E16">C4</font></i>: 
( <font color="Maroon" title="c13">Y2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon" title="c2">seq2</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Olive" title="b1">k</font></span>)</span> where <font color="Olive" title="b1">k</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>  : <font color="Maroon" title="c12">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">k</font> </span>}</span>  &amp; <font color="Maroon" title="c10">z3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_2.html#K7" title="SUPINF_2:func.7">inf</a> <font color="Maroon" title="c13">Y2</font> )
 <b>by </b><i><a class="txt" href="#E5:10_2"><i><font color="Green" title="E15">C3</font></i></a>, <a class="ref" href="rinfsup2.html#D6" title="RINFSUP2:def.6">RINFSUP2:def 6</a></i>;<br/><b>consider </b><font color="Maroon" title="c14">Y1</font> being  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <a href="numbers.html#K7" title="NUMBERS:func.7">ExtREAL</a> <b> such that </b><br/><a NAME="E10:10_2"/><i><font color="Green" title="E17">C5</font></i>: 
( <font color="Maroon" title="c14">Y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon" title="c1">seq1</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Olive" title="b1">k</font></span>)</span> where <font color="Olive" title="b1">k</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>  : <font color="Maroon" title="c12">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">k</font> </span>}</span>  &amp; <span class="p1">(<span class="default"><a href="rinfsup2.html#K3" title="RINFSUP2:func.3">inferior_realsequence</a> <font color="Maroon" title="c1">seq1</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>  <a href="supinf_2.html#K7" title="SUPINF_2:func.7">inf</a> <font color="Maroon" title="c14">Y1</font> )
 <b>by </b><i><a class="ref" href="rinfsup2.html#D6" title="RINFSUP2:def.6">RINFSUP2:def 6</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="10_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c15">w</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c15">w</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c14">Y1</font> implies <font color="Maroon" title="c15">w</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c3">r</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="supinf_2.html#K5" title="SUPINF_2:func.5">+</a> <font color="Maroon" title="c13">Y2</font> )</span><br/><b>assume </b><a NAME="E1:10_2_1"/>
<font color="Maroon" title="c15">w</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c14">Y1</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c15">w</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c3">r</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="supinf_2.html#K5" title="SUPINF_2:func.5">+</a> <font color="Maroon" title="c13">Y2</font></span><br/><b>then consider </b><font color="Maroon" title="c16">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> <b> such that </b><br/><a NAME="E3:10_2_1"/><i><font color="Green" title="E18">B3</font></i>: 
( <font color="Maroon" title="c15">w</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> <font color="Maroon" title="c16">k1</font> &amp; <font color="Maroon" title="c12">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c16">k1</font> )
 <b>by </b><i><a class="txt" href="#E10:10_2"><i><font color="Green" title="E17">C5</font></i></a></i>;<br/><b>reconsider </b><font color="Maroon" title="c17">w1</font> = <font color="Maroon" title="c15">w</font> as   <a href="supinf_1.html#NM3" title="SUPINF_1:NM.3">R_eal</a> <b>by </b><i><a class="txt" href="#E3:10_2_1"><i><font color="Green" title="E18">B3</font></i></a></i>;<br/><a NAME="E5:10_2_1"/><i><font color="Green" title="E19">B4</font></i>: 
<font color="Maroon" title="c17">w1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">r</font> <a href="supinf_2.html#K2" title="SUPINF_2:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">seq2</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c16">k1</font></span>)</span>
 <b>by </b><i><a class="txt" href="#E2:10"><i><font color="Green" title="E12">A1</font></i></a>, <a class="txt" href="#E3:10_2_1"><i><font color="Green" title="E18">B3</font></i></a></i>;<br/><a NAME="E6:10_2_1"/>
( <font color="Maroon" title="c3">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c3">r</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> &amp; <font color="Maroon" title="c2">seq2</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c16">k1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c13">Y2</font> )
 <b>by </b><i><a class="txt" href="#E3:10_2_1"><i><font color="Green" title="E18">B3</font></i></a>, <a class="txt" href="#E8:10_2"><i><font color="Green" title="E16">C4</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/><b>hence </b><a NAME="E7:10_2_1"/>
<font color="Maroon" title="c15">w</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c3">r</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="supinf_2.html#K5" title="SUPINF_2:func.5">+</a> <font color="Maroon" title="c13">Y2</font>
 <b>by </b><i><a class="txt" href="#E5:10_2_1"><i><font color="Green" title="E19">B4</font></i></a>, <a class="ref" href="supinf_2.html#D5" title="SUPINF_2:def.5">SUPINF_2:def 5</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><a NAME="E12:10_2"/><b>then </b><i><font color="Green" title="E18">B6</font></i>: 
<font color="Maroon" title="c14">Y1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c3">r</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="supinf_2.html#K5" title="SUPINF_2:func.5">+</a> <font color="Maroon" title="c13">Y2</font>
 <b>by </b><i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="10_2_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c15">w</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c15">w</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c3">r</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="supinf_2.html#K5" title="SUPINF_2:func.5">+</a> <font color="Maroon" title="c13">Y2</font> implies <font color="Maroon" title="c15">w</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c14">Y1</font> )</span><br/><b>assume </b><a NAME="E1:10_2_2"/>
<font color="Maroon" title="c15">w</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c3">r</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="supinf_2.html#K5" title="SUPINF_2:func.5">+</a> <font color="Maroon" title="c13">Y2</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c15">w</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c14">Y1</font></span><br/><b>then consider </b><font color="Maroon" title="c16">w1</font>, <font color="Maroon" title="c17">w2</font> being   <a href="supinf_1.html#NM3" title="SUPINF_1:NM.3">R_eal</a><b> such that </b><br/><a NAME="E3:10_2_2"/><i><font color="Green" title="E19">B7</font></i>: 
( <font color="Maroon" title="c16">w1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c3">r</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> &amp; <font color="Maroon" title="c17">w2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c13">Y2</font> &amp; <font color="Maroon" title="c15">w</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">w1</font> <a href="supinf_2.html#K2" title="SUPINF_2:func.2">+</a> <font color="Maroon" title="c17">w2</font> )
 <b>by </b><i><a class="ref" href="supinf_2.html#D5" title="SUPINF_2:def.5">SUPINF_2:def 5</a></i>;<br/><b>consider </b><font color="Maroon" title="c18">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> <b> such that </b><br/><a NAME="E5:10_2_2"/><i><font color="Green" title="E20">B8</font></i>: 
( <font color="Maroon" title="c17">w2</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> <font color="Maroon" title="c18">k2</font> &amp; <font color="Maroon" title="c12">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c18">k2</font> )
 <b>by </b><i><a class="txt" href="#E8:10_2"><i><font color="Green" title="E16">C4</font></i></a>, <a class="txt" href="#E3:10_2_2"><i><font color="Green" title="E19">B7</font></i></a></i>;<br/><a NAME="E6:10_2_2"/>
<font color="Maroon" title="c16">w1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">r</font>
 <b>by </b><i><a class="txt" href="#E3:10_2_2"><i><font color="Green" title="E19">B7</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/><a NAME="E7:10_2_2"/><b>then </b>
<font color="Maroon" title="c15">w</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> <font color="Maroon" title="c18">k2</font>
 <b>by </b><i><a class="txt" href="#E2:10"><i><font color="Green" title="E12">A1</font></i></a>, <a class="txt" href="#E3:10_2_2"><i><font color="Green" title="E19">B7</font></i></a>, <a class="txt" href="#E5:10_2_2"><i><font color="Green" title="E20">B8</font></i></a></i>;<br/><b>hence </b><a NAME="E8:10_2_2"/>
<font color="Maroon" title="c15">w</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c14">Y1</font>
 <b>by </b><i><a class="txt" href="#E10:10_2"><i><font color="Green" title="E17">C5</font></i></a>, <a class="txt" href="#E5:10_2_2"><i><font color="Green" title="E20">B8</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><a NAME="E14:10_2"/><b>then </b>
<span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c3">r</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="supinf_2.html#K5" title="SUPINF_2:func.5">+</a> <font color="Maroon" title="c13">Y2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c14">Y1</font>
 <b>by </b><i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></i>;<br/><a NAME="E15:10_2"/><b>then </b>
<font color="Maroon" title="c14">Y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c3">r</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="supinf_2.html#K5" title="SUPINF_2:func.5">+</a> <font color="Maroon" title="c13">Y2</font>
 <b>by </b><i><a class="txt" href="#E12:10_2"><i><font color="Green" title="E18">B6</font></i></a>, <a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a></i>;<br/><a NAME="E16:10_2"/><b>then </b>
 <a href="supinf_2.html#K7" title="SUPINF_2:func.7">inf</a> <font color="Maroon" title="c14">Y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K7" title="SUPINF_2:func.7">inf</a> <span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c3">r</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span> <a href="supinf_2.html#K2" title="SUPINF_2:func.2">+</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K7" title="SUPINF_2:func.7">inf</a> <font color="Maroon" title="c13">Y2</font></span>)</span>
 <b>by </b><i><a class="txt" href="#E1:10"><i><font color="Green" title="E11">A0</font></i></a>, <a class="ref" href="mesfun10.html#T9" target="_self" title="MESFUN10:th.9">SUPINF227a</a></i>;<br/><a NAME="E17:10_2"/><b>then </b>
 <a href="supinf_2.html#K7" title="SUPINF_2:func.7">inf</a> <font color="Maroon" title="c14">Y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">r</font> <a href="supinf_2.html#K2" title="SUPINF_2:func.2">+</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K7" title="SUPINF_2:func.7">inf</a> <font color="Maroon" title="c13">Y2</font></span>)</span>
 <b>by </b><i><a class="ref" href="xxreal_2.html#T13" title="XXREAL_2:th.13">XXREAL_2:13</a></i>;<br/><a NAME="E18:10_2"/><b>then </b>
<font color="Maroon" title="c8">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="rinfsup2.html#K3" title="RINFSUP2:func.3">inferior_realsequence</a> <font color="Maroon" title="c1">seq1</font></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c12">n</font>
 <b>by </b><i><a class="txt" href="#E3:10_2"><i><font color="Green" title="E14">C2</font></i></a>, <a class="txt" href="#E8:10_2"><i><font color="Green" title="E16">C4</font></i></a>, <a class="txt" href="#E10:10_2"><i><font color="Green" title="E17">C5</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/><b>hence </b><a NAME="E19:10_2"/>
<font color="Maroon" title="c8">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">R1</font>
 <b>by </b><i><a class="ref" href="funct_2.html#T6" title="FUNCT_2:th.6">FUNCT_2:6</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div>
<a NAME="E7:10"/><b>then </b>
<span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c3">r</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="supinf_2.html#K5" title="SUPINF_2:func.5">+</a> <font color="Maroon" title="c5">R2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">R1</font>
 
<b>by </b><i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></i>;<br/>
<a NAME="E8:10"/><b>then </b>
<font color="Maroon" title="c4">R1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c3">r</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="supinf_2.html#K5" title="SUPINF_2:func.5">+</a> <font color="Maroon" title="c5">R2</font>
 
<b>by </b><i><a class="txt" href="#E5:10"><i><font color="Green" title="E13">B9</font></i></a>, <a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a></i>;<br/>
<a NAME="E9:10"/><b>then </b>
 <a href="supinf_2.html#K8" title="SUPINF_2:func.8">sup</a> <font color="Maroon" title="c4">R1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K8" title="SUPINF_2:func.8">sup</a> <span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c3">r</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span> <a href="supinf_2.html#K2" title="SUPINF_2:func.2">+</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K8" title="SUPINF_2:func.8">sup</a> <font color="Maroon" title="c5">R2</font></span>)</span>
 
<b>by </b><i><a class="txt" href="#E1:10"><i><font color="Green" title="E11">A0</font></i></a>, <a class="ref" href="mesfun10.html#T8" target="_self" title="MESFUN10:th.8">SUPINF226a</a></i>;<br/>
<b>hence </b><a NAME="E10:10"/>
 <a href="rinfsup2.html#K6" title="RINFSUP2:func.6">lim_inf</a> <font color="Maroon" title="c1">seq1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">r</font> <a href="supinf_2.html#K2" title="SUPINF_2:func.2">+</a> <span class="p1">(<span class="default"><a href="rinfsup2.html#K6" title="RINFSUP2:func.6">lim_inf</a> <font color="Maroon" title="c2">seq2</font></span>)</span>
 <b>by </b><i><a class="ref" href="xxreal_2.html#T11" title="XXREAL_2:th.11">XXREAL_2:11</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="rinfsup2.html#K5" title="RINFSUP2:func.5">lim_sup</a> <font color="Maroon" title="c1">seq1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">r</font> <a href="supinf_2.html#K2" title="SUPINF_2:func.2">+</a> <span class="p1">(<span class="default"><a href="rinfsup2.html#K5" title="RINFSUP2:func.5">lim_sup</a> <font color="Maroon" title="c2">seq2</font></span>)</span></span><br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="10_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c8">z</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c8">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">P1</font> implies <font color="Maroon" title="c8">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c3">r</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="supinf_2.html#K5" title="SUPINF_2:func.5">+</a> <font color="Maroon" title="c7">P2</font> )</span><br/><b>assume </b><a NAME="E1:10_3"/>
<font color="Maroon" title="c8">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">P1</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c8">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c3">r</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="supinf_2.html#K5" title="SUPINF_2:func.5">+</a> <font color="Maroon" title="c7">P2</font></span><br/><b>then consider </b><font color="Maroon" title="c9">n</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:10_3"/><i><font color="Green" title="E14">B1</font></i>: 
( <font color="Maroon" title="c9">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  &amp; <span class="p1">(<span class="default"><a href="rinfsup2.html#K4" title="RINFSUP2:func.4">superior_realsequence</a> <font color="Maroon" title="c1">seq1</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">z</font> )
 <b>by </b><i><a class="ref" href="funct_2.html#T17" title="FUNCT_2:th.17">FUNCT_2:17</a></i>;<br/><b>reconsider </b><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>  <b>by </b><i><a class="txt" href="#E3:10_3"><i><font color="Green" title="E14">B1</font></i></a></i>;<br/><b>consider </b><font color="Maroon" title="c11">Y1</font> being  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <a href="numbers.html#K7" title="NUMBERS:func.7">ExtREAL</a> <b> such that </b><br/><a NAME="E6:10_3"/><i><font color="Green" title="E15">B2</font></i>: 
( <font color="Maroon" title="c11">Y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon" title="c1">seq1</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Olive" title="b1">k</font></span>)</span> where <font color="Olive" title="b1">k</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>  : <font color="Maroon" title="c10">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">k</font> </span>}</span>  &amp; <font color="Maroon" title="c8">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_2.html#K8" title="SUPINF_2:func.8">sup</a> <font color="Maroon" title="c11">Y1</font> )
 <b>by </b><i><a class="txt" href="#E3:10_3"><i><font color="Green" title="E14">B1</font></i></a>, <a class="ref" href="rinfsup2.html#D7" title="RINFSUP2:def.7">RINFSUP2:def 7</a></i>;<br/><b>consider </b><font color="Maroon" title="c12">Y2</font> being  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <a href="numbers.html#K7" title="NUMBERS:func.7">ExtREAL</a> <b> such that </b><br/><a NAME="E8:10_3"/><i><font color="Green" title="E16">B21</font></i>: 
( <font color="Maroon" title="c12">Y2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon" title="c2">seq2</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Olive" title="b1">k</font></span>)</span> where <font color="Olive" title="b1">k</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>  : <font color="Maroon" title="c10">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">k</font> </span>}</span>  &amp; <span class="p1">(<span class="default"><a href="rinfsup2.html#K4" title="RINFSUP2:func.4">superior_realsequence</a> <font color="Maroon" title="c2">seq2</font></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c10">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_2.html#K8" title="SUPINF_2:func.8">sup</a> <font color="Maroon" title="c12">Y2</font> )
 <b>by </b><i><a class="ref" href="rinfsup2.html#D7" title="RINFSUP2:def.7">RINFSUP2:def 7</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="10_3_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c13">w</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c13">w</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">Y1</font> implies <font color="Maroon" title="c13">w</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c3">r</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="supinf_2.html#K5" title="SUPINF_2:func.5">+</a> <font color="Maroon" title="c12">Y2</font> )</span><br/><b>assume </b><a NAME="E1:10_3_1"/>
<font color="Maroon" title="c13">w</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">Y1</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c13">w</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c3">r</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="supinf_2.html#K5" title="SUPINF_2:func.5">+</a> <font color="Maroon" title="c12">Y2</font></span><br/><b>then consider </b><font color="Maroon" title="c14">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> <b> such that </b><br/><a NAME="E3:10_3_1"/><i><font color="Green" title="E17">B3</font></i>: 
( <font color="Maroon" title="c13">w</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> <font color="Maroon" title="c14">k1</font> &amp; <font color="Maroon" title="c10">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c14">k1</font> )
 <b>by </b><i><a class="txt" href="#E6:10_3"><i><font color="Green" title="E15">B2</font></i></a></i>;<br/><b>reconsider </b><font color="Maroon" title="c15">w1</font> = <font color="Maroon" title="c13">w</font> as   <a href="supinf_1.html#NM3" title="SUPINF_1:NM.3">R_eal</a> <b>by </b><i><a class="txt" href="#E3:10_3_1"><i><font color="Green" title="E17">B3</font></i></a></i>;<br/><a NAME="E5:10_3_1"/><i><font color="Green" title="E18">B4</font></i>: 
<font color="Maroon" title="c15">w1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">r</font> <a href="supinf_2.html#K2" title="SUPINF_2:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">seq2</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c14">k1</font></span>)</span>
 <b>by </b><i><a class="txt" href="#E2:10"><i><font color="Green" title="E12">A1</font></i></a>, <a class="txt" href="#E3:10_3_1"><i><font color="Green" title="E17">B3</font></i></a></i>;<br/><a NAME="E6:10_3_1"/>
( <font color="Maroon" title="c3">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c3">r</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> &amp; <font color="Maroon" title="c2">seq2</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c14">k1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c12">Y2</font> )
 <b>by </b><i><a class="txt" href="#E3:10_3_1"><i><font color="Green" title="E17">B3</font></i></a>, <a class="txt" href="#E8:10_3"><i><font color="Green" title="E16">B21</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/><b>hence </b><a NAME="E7:10_3_1"/>
<font color="Maroon" title="c13">w</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c3">r</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="supinf_2.html#K5" title="SUPINF_2:func.5">+</a> <font color="Maroon" title="c12">Y2</font>
 <b>by </b><i><a class="txt" href="#E5:10_3_1"><i><font color="Green" title="E18">B4</font></i></a>, <a class="ref" href="supinf_2.html#D5" title="SUPINF_2:def.5">SUPINF_2:def 5</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><a NAME="E10:10_3"/><b>then </b><i><font color="Green" title="E17">B6</font></i>: 
<font color="Maroon" title="c11">Y1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c3">r</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="supinf_2.html#K5" title="SUPINF_2:func.5">+</a> <font color="Maroon" title="c12">Y2</font>
 <b>by </b><i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="10_3_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c13">w</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c13">w</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c3">r</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="supinf_2.html#K5" title="SUPINF_2:func.5">+</a> <font color="Maroon" title="c12">Y2</font> implies <font color="Maroon" title="c13">w</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">Y1</font> )</span><br/><b>assume </b><a NAME="E1:10_3_2"/>
<font color="Maroon" title="c13">w</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c3">r</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="supinf_2.html#K5" title="SUPINF_2:func.5">+</a> <font color="Maroon" title="c12">Y2</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c13">w</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">Y1</font></span><br/><b>then consider </b><font color="Maroon" title="c14">w1</font>, <font color="Maroon" title="c15">w2</font> being   <a href="supinf_1.html#NM3" title="SUPINF_1:NM.3">R_eal</a><b> such that </b><br/><a NAME="E3:10_3_2"/><i><font color="Green" title="E18">B7</font></i>: 
( <font color="Maroon" title="c14">w1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c3">r</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> &amp; <font color="Maroon" title="c15">w2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c12">Y2</font> &amp; <font color="Maroon" title="c13">w</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">w1</font> <a href="supinf_2.html#K2" title="SUPINF_2:func.2">+</a> <font color="Maroon" title="c15">w2</font> )
 <b>by </b><i><a class="ref" href="supinf_2.html#D5" title="SUPINF_2:def.5">SUPINF_2:def 5</a></i>;<br/><b>consider </b><font color="Maroon" title="c16">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> <b> such that </b><br/><a NAME="E5:10_3_2"/><i><font color="Green" title="E19">B8</font></i>: 
( <font color="Maroon" title="c15">w2</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> <font color="Maroon" title="c16">k2</font> &amp; <font color="Maroon" title="c10">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c16">k2</font> )
 <b>by </b><i><a class="txt" href="#E8:10_3"><i><font color="Green" title="E16">B21</font></i></a>, <a class="txt" href="#E3:10_3_2"><i><font color="Green" title="E18">B7</font></i></a></i>;<br/><a NAME="E6:10_3_2"/>
<font color="Maroon" title="c14">w1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">r</font>
 <b>by </b><i><a class="txt" href="#E3:10_3_2"><i><font color="Green" title="E18">B7</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/><a NAME="E7:10_3_2"/><b>then </b>
<font color="Maroon" title="c13">w</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> <font color="Maroon" title="c16">k2</font>
 <b>by </b><i><a class="txt" href="#E2:10"><i><font color="Green" title="E12">A1</font></i></a>, <a class="txt" href="#E3:10_3_2"><i><font color="Green" title="E18">B7</font></i></a>, <a class="txt" href="#E5:10_3_2"><i><font color="Green" title="E19">B8</font></i></a></i>;<br/><b>hence </b><a NAME="E8:10_3_2"/>
<font color="Maroon" title="c13">w</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">Y1</font>
 <b>by </b><i><a class="txt" href="#E6:10_3"><i><font color="Green" title="E15">B2</font></i></a>, <a class="txt" href="#E5:10_3_2"><i><font color="Green" title="E19">B8</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><a NAME="E12:10_3"/><b>then </b>
<span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c3">r</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="supinf_2.html#K5" title="SUPINF_2:func.5">+</a> <font color="Maroon" title="c12">Y2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c11">Y1</font>
 <b>by </b><i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></i>;<br/><a NAME="E13:10_3"/><b>then </b>
<font color="Maroon" title="c11">Y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c3">r</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="supinf_2.html#K5" title="SUPINF_2:func.5">+</a> <font color="Maroon" title="c12">Y2</font>
 <b>by </b><i><a class="txt" href="#E10:10_3"><i><font color="Green" title="E17">B6</font></i></a>, <a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a></i>;<br/><a NAME="E14:10_3"/><b>then </b>
 <a href="supinf_2.html#K8" title="SUPINF_2:func.8">sup</a> <font color="Maroon" title="c11">Y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K8" title="SUPINF_2:func.8">sup</a> <span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c3">r</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span> <a href="supinf_2.html#K2" title="SUPINF_2:func.2">+</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K8" title="SUPINF_2:func.8">sup</a> <font color="Maroon" title="c12">Y2</font></span>)</span>
 <b>by </b><i><a class="txt" href="#E1:10"><i><font color="Green" title="E11">A0</font></i></a>, <a class="ref" href="mesfun10.html#T8" target="_self" title="MESFUN10:th.8">SUPINF226a</a></i>;<br/><a NAME="E15:10_3"/><b>then </b><i><font color="Green" title="E18">B9</font></i>: 
 <a href="supinf_2.html#K8" title="SUPINF_2:func.8">sup</a> <font color="Maroon" title="c11">Y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">r</font> <a href="supinf_2.html#K2" title="SUPINF_2:func.2">+</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K8" title="SUPINF_2:func.8">sup</a> <font color="Maroon" title="c12">Y2</font></span>)</span>
 <b>by </b><i><a class="ref" href="xxreal_2.html#T11" title="XXREAL_2:th.11">XXREAL_2:11</a></i>;<br/><a NAME="E16:10_3"/>
( <font color="Maroon" title="c3">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c3">r</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> &amp; <span class="p1">(<span class="default"><a href="rinfsup2.html#K4" title="RINFSUP2:func.4">superior_realsequence</a> <font color="Maroon" title="c2">seq2</font></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c10">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">P2</font> )
 <b>by </b><i><a class="ref" href="funct_2.html#T6" title="FUNCT_2:th.6">FUNCT_2:6</a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/><b>hence </b><a NAME="E17:10_3"/>
<font color="Maroon" title="c8">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c3">r</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="supinf_2.html#K5" title="SUPINF_2:func.5">+</a> <font color="Maroon" title="c7">P2</font>
 <b>by </b><i><a class="txt" href="#E15:10_3"><i><font color="Green" title="E18">B9</font></i></a>, <a class="txt" href="#E6:10_3"><i><font color="Green" title="E15">B2</font></i></a>, <a class="txt" href="#E8:10_3"><i><font color="Green" title="E16">B21</font></i></a>, <a class="ref" href="supinf_2.html#D5" title="SUPINF_2:def.5">SUPINF_2:def 5</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div>
<a NAME="E12:10"/><b>then </b><i><font color="Green" title="E14">B9</font></i>: 
<font color="Maroon" title="c6">P1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c3">r</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="supinf_2.html#K5" title="SUPINF_2:func.5">+</a> <font color="Maroon" title="c7">P2</font>
 
<b>by </b><i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="10_4"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c8">z</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c8">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c3">r</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="supinf_2.html#K5" title="SUPINF_2:func.5">+</a> <font color="Maroon" title="c7">P2</font> implies <font color="Maroon" title="c8">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">P1</font> )</span><br/><b>assume </b><a NAME="E1:10_4"/>
<font color="Maroon" title="c8">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c3">r</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="supinf_2.html#K5" title="SUPINF_2:func.5">+</a> <font color="Maroon" title="c7">P2</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c8">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">P1</font></span><br/><b>then consider </b><font color="Maroon" title="c9">z2</font>, <font color="Maroon" title="c10">z3</font> being   <a href="supinf_1.html#NM3" title="SUPINF_1:NM.3">R_eal</a><b> such that </b><br/><a NAME="E3:10_4"/><i><font color="Green" title="E15">C2</font></i>: 
( <font color="Maroon" title="c9">z2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c3">r</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> &amp; <font color="Maroon" title="c10">z3</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">P2</font> &amp; <font color="Maroon" title="c8">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">z2</font> <a href="supinf_2.html#K2" title="SUPINF_2:func.2">+</a> <font color="Maroon" title="c10">z3</font> )
 <b>by </b><i><a class="ref" href="supinf_2.html#D5" title="SUPINF_2:def.5">SUPINF_2:def 5</a></i>;<br/><b>consider </b><font color="Maroon" title="c11">n</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E5:10_4"/><i><font color="Green" title="E16">C3</font></i>: 
( <font color="Maroon" title="c11">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  &amp; <span class="p1">(<span class="default"><a href="rinfsup2.html#K4" title="RINFSUP2:func.4">superior_realsequence</a> <font color="Maroon" title="c2">seq2</font></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c11">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">z3</font> )
 <b>by </b><i><a class="txt" href="#E3:10_4"><i><font color="Green" title="E15">C2</font></i></a>, <a class="ref" href="funct_2.html#T17" title="FUNCT_2:th.17">FUNCT_2:17</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c12">n</font> = <font color="Maroon" title="c11">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>  <b>by </b><i><a class="txt" href="#E5:10_4"><i><font color="Green" title="E16">C3</font></i></a></i>;<br/><b>consider </b><font color="Maroon" title="c13">Y2</font> being  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <a href="numbers.html#K7" title="NUMBERS:func.7">ExtREAL</a> <b> such that </b><br/><a NAME="E8:10_4"/><i><font color="Green" title="E17">C4</font></i>: 
( <font color="Maroon" title="c13">Y2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon" title="c2">seq2</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Olive" title="b1">k</font></span>)</span> where <font color="Olive" title="b1">k</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>  : <font color="Maroon" title="c12">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">k</font> </span>}</span>  &amp; <font color="Maroon" title="c10">z3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_2.html#K8" title="SUPINF_2:func.8">sup</a> <font color="Maroon" title="c13">Y2</font> )
 <b>by </b><i><a class="txt" href="#E5:10_4"><i><font color="Green" title="E16">C3</font></i></a>, <a class="ref" href="rinfsup2.html#D7" title="RINFSUP2:def.7">RINFSUP2:def 7</a></i>;<br/><b>consider </b><font color="Maroon" title="c14">Y1</font> being  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <a href="numbers.html#K7" title="NUMBERS:func.7">ExtREAL</a> <b> such that </b><br/><a NAME="E10:10_4"/><i><font color="Green" title="E18">C5</font></i>: 
( <font color="Maroon" title="c14">Y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon" title="c1">seq1</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Olive" title="b1">k</font></span>)</span> where <font color="Olive" title="b1">k</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>  : <font color="Maroon" title="c12">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">k</font> </span>}</span>  &amp; <span class="p1">(<span class="default"><a href="rinfsup2.html#K4" title="RINFSUP2:func.4">superior_realsequence</a> <font color="Maroon" title="c1">seq1</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>  <a href="supinf_2.html#K8" title="SUPINF_2:func.8">sup</a> <font color="Maroon" title="c14">Y1</font> )
 <b>by </b><i><a class="ref" href="rinfsup2.html#D7" title="RINFSUP2:def.7">RINFSUP2:def 7</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="10_4_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c15">w</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c15">w</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c14">Y1</font> implies <font color="Maroon" title="c15">w</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c3">r</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="supinf_2.html#K5" title="SUPINF_2:func.5">+</a> <font color="Maroon" title="c13">Y2</font> )</span><br/><b>assume </b><a NAME="E1:10_4_1"/>
<font color="Maroon" title="c15">w</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c14">Y1</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c15">w</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c3">r</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="supinf_2.html#K5" title="SUPINF_2:func.5">+</a> <font color="Maroon" title="c13">Y2</font></span><br/><b>then consider </b><font color="Maroon" title="c16">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> <b> such that </b><br/><a NAME="E3:10_4_1"/><i><font color="Green" title="E19">B3</font></i>: 
( <font color="Maroon" title="c15">w</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> <font color="Maroon" title="c16">k1</font> &amp; <font color="Maroon" title="c12">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c16">k1</font> )
 <b>by </b><i><a class="txt" href="#E10:10_4"><i><font color="Green" title="E18">C5</font></i></a></i>;<br/><b>reconsider </b><font color="Maroon" title="c17">w1</font> = <font color="Maroon" title="c15">w</font> as   <a href="supinf_1.html#NM3" title="SUPINF_1:NM.3">R_eal</a> <b>by </b><i><a class="txt" href="#E3:10_4_1"><i><font color="Green" title="E19">B3</font></i></a></i>;<br/><a NAME="E5:10_4_1"/><i><font color="Green" title="E20">B4</font></i>: 
<font color="Maroon" title="c17">w1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">r</font> <a href="supinf_2.html#K2" title="SUPINF_2:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">seq2</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c16">k1</font></span>)</span>
 <b>by </b><i><a class="txt" href="#E2:10"><i><font color="Green" title="E12">A1</font></i></a>, <a class="txt" href="#E3:10_4_1"><i><font color="Green" title="E19">B3</font></i></a></i>;<br/><a NAME="E6:10_4_1"/>
( <font color="Maroon" title="c3">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c3">r</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> &amp; <font color="Maroon" title="c2">seq2</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c16">k1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c13">Y2</font> )
 <b>by </b><i><a class="txt" href="#E3:10_4_1"><i><font color="Green" title="E19">B3</font></i></a>, <a class="txt" href="#E8:10_4"><i><font color="Green" title="E17">C4</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/><b>hence </b><a NAME="E7:10_4_1"/>
<font color="Maroon" title="c15">w</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c3">r</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="supinf_2.html#K5" title="SUPINF_2:func.5">+</a> <font color="Maroon" title="c13">Y2</font>
 <b>by </b><i><a class="txt" href="#E5:10_4_1"><i><font color="Green" title="E20">B4</font></i></a>, <a class="ref" href="supinf_2.html#D5" title="SUPINF_2:def.5">SUPINF_2:def 5</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><a NAME="E12:10_4"/><b>then </b><i><font color="Green" title="E19">B6</font></i>: 
<font color="Maroon" title="c14">Y1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c3">r</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="supinf_2.html#K5" title="SUPINF_2:func.5">+</a> <font color="Maroon" title="c13">Y2</font>
 <b>by </b><i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="10_4_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c15">w</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c15">w</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c3">r</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="supinf_2.html#K5" title="SUPINF_2:func.5">+</a> <font color="Maroon" title="c13">Y2</font> implies <font color="Maroon" title="c15">w</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c14">Y1</font> )</span><br/><b>assume </b><a NAME="E1:10_4_2"/>
<font color="Maroon" title="c15">w</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c3">r</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="supinf_2.html#K5" title="SUPINF_2:func.5">+</a> <font color="Maroon" title="c13">Y2</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c15">w</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c14">Y1</font></span><br/><b>then consider </b><font color="Maroon" title="c16">w1</font>, <font color="Maroon" title="c17">w2</font> being   <a href="supinf_1.html#NM3" title="SUPINF_1:NM.3">R_eal</a><b> such that </b><br/><a NAME="E3:10_4_2"/><i><font color="Green" title="E20">B7</font></i>: 
( <font color="Maroon" title="c16">w1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c3">r</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> &amp; <font color="Maroon" title="c17">w2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c13">Y2</font> &amp; <font color="Maroon" title="c15">w</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">w1</font> <a href="supinf_2.html#K2" title="SUPINF_2:func.2">+</a> <font color="Maroon" title="c17">w2</font> )
 <b>by </b><i><a class="ref" href="supinf_2.html#D5" title="SUPINF_2:def.5">SUPINF_2:def 5</a></i>;<br/><b>consider </b><font color="Maroon" title="c18">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> <b> such that </b><br/><a NAME="E5:10_4_2"/><i><font color="Green" title="E21">B8</font></i>: 
( <font color="Maroon" title="c17">w2</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> <font color="Maroon" title="c18">k2</font> &amp; <font color="Maroon" title="c12">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c18">k2</font> )
 <b>by </b><i><a class="txt" href="#E8:10_4"><i><font color="Green" title="E17">C4</font></i></a>, <a class="txt" href="#E3:10_4_2"><i><font color="Green" title="E20">B7</font></i></a></i>;<br/><a NAME="E6:10_4_2"/>
<font color="Maroon" title="c16">w1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">r</font>
 <b>by </b><i><a class="txt" href="#E3:10_4_2"><i><font color="Green" title="E20">B7</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/><a NAME="E7:10_4_2"/><b>then </b>
<font color="Maroon" title="c15">w</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> <font color="Maroon" title="c18">k2</font>
 <b>by </b><i><a class="txt" href="#E2:10"><i><font color="Green" title="E12">A1</font></i></a>, <a class="txt" href="#E3:10_4_2"><i><font color="Green" title="E20">B7</font></i></a>, <a class="txt" href="#E5:10_4_2"><i><font color="Green" title="E21">B8</font></i></a></i>;<br/><b>hence </b><a NAME="E8:10_4_2"/>
<font color="Maroon" title="c15">w</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c14">Y1</font>
 <b>by </b><i><a class="txt" href="#E10:10_4"><i><font color="Green" title="E18">C5</font></i></a>, <a class="txt" href="#E5:10_4_2"><i><font color="Green" title="E21">B8</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><a NAME="E14:10_4"/><b>then </b>
<span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c3">r</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="supinf_2.html#K5" title="SUPINF_2:func.5">+</a> <font color="Maroon" title="c13">Y2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c14">Y1</font>
 <b>by </b><i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></i>;<br/><a NAME="E15:10_4"/><b>then </b>
<font color="Maroon" title="c14">Y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c3">r</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="supinf_2.html#K5" title="SUPINF_2:func.5">+</a> <font color="Maroon" title="c13">Y2</font>
 <b>by </b><i><a class="txt" href="#E12:10_4"><i><font color="Green" title="E19">B6</font></i></a>, <a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a></i>;<br/><a NAME="E16:10_4"/><b>then </b>
 <a href="supinf_2.html#K8" title="SUPINF_2:func.8">sup</a> <font color="Maroon" title="c14">Y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K8" title="SUPINF_2:func.8">sup</a> <span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c3">r</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span> <a href="supinf_2.html#K2" title="SUPINF_2:func.2">+</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K8" title="SUPINF_2:func.8">sup</a> <font color="Maroon" title="c13">Y2</font></span>)</span>
 <b>by </b><i><a class="txt" href="#E1:10"><i><font color="Green" title="E11">A0</font></i></a>, <a class="ref" href="mesfun10.html#T8" target="_self" title="MESFUN10:th.8">SUPINF226a</a></i>;<br/><a NAME="E17:10_4"/><b>then </b>
 <a href="supinf_2.html#K8" title="SUPINF_2:func.8">sup</a> <font color="Maroon" title="c14">Y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">r</font> <a href="supinf_2.html#K2" title="SUPINF_2:func.2">+</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K8" title="SUPINF_2:func.8">sup</a> <font color="Maroon" title="c13">Y2</font></span>)</span>
 <b>by </b><i><a class="ref" href="xxreal_2.html#T11" title="XXREAL_2:th.11">XXREAL_2:11</a></i>;<br/><a NAME="E18:10_4"/><b>then </b>
<font color="Maroon" title="c8">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="rinfsup2.html#K4" title="RINFSUP2:func.4">superior_realsequence</a> <font color="Maroon" title="c1">seq1</font></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c12">n</font>
 <b>by </b><i><a class="txt" href="#E3:10_4"><i><font color="Green" title="E15">C2</font></i></a>, <a class="txt" href="#E8:10_4"><i><font color="Green" title="E17">C4</font></i></a>, <a class="txt" href="#E10:10_4"><i><font color="Green" title="E18">C5</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/><b>hence </b><a NAME="E19:10_4"/>
<font color="Maroon" title="c8">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">P1</font>
 <b>by </b><i><a class="ref" href="funct_2.html#T6" title="FUNCT_2:th.6">FUNCT_2:6</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div>
<a NAME="E14:10"/><b>then </b>
<span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c3">r</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="supinf_2.html#K5" title="SUPINF_2:func.5">+</a> <font color="Maroon" title="c7">P2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c6">P1</font>
 
<b>by </b><i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></i>;<br/>
<a NAME="E15:10"/><b>then </b>
<font color="Maroon" title="c6">P1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c3">r</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="supinf_2.html#K5" title="SUPINF_2:func.5">+</a> <font color="Maroon" title="c7">P2</font>
 
<b>by </b><i><a class="txt" href="#E12:10"><i><font color="Green" title="E14">B9</font></i></a>, <a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a></i>;<br/>
<a NAME="E16:10"/><b>then </b>
 <a href="supinf_2.html#K7" title="SUPINF_2:func.7">inf</a> <font color="Maroon" title="c6">P1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K7" title="SUPINF_2:func.7">inf</a> <span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c3">r</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span> <a href="supinf_2.html#K2" title="SUPINF_2:func.2">+</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K7" title="SUPINF_2:func.7">inf</a> <font color="Maroon" title="c7">P2</font></span>)</span>
 
<b>by </b><i><a class="txt" href="#E1:10"><i><font color="Green" title="E11">A0</font></i></a>, <a class="ref" href="mesfun10.html#T9" target="_self" title="MESFUN10:th.9">SUPINF227a</a></i>;<br/>
<b>hence </b><a NAME="E17:10"/>
 <a href="rinfsup2.html#K5" title="RINFSUP2:func.5">lim_sup</a> <font color="Maroon" title="c1">seq1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">r</font> <a href="supinf_2.html#K2" title="SUPINF_2:func.2">+</a> <span class="p1">(<span class="default"><a href="rinfsup2.html#K5" title="RINFSUP2:func.5">lim_sup</a> <font color="Maroon" title="c2">seq2</font></span>)</span>
 <b>by </b><i><a class="ref" href="xxreal_2.html#T13" title="XXREAL_2:th.13">XXREAL_2:13</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div>
