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

<b>let </b><font color="Maroon" title="c1">T</font> be  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> <a href="pre_topc.html#NM1" title="PRE_TOPC:NM.1">TopSpace</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c1">T</font> is <a href="frechet.html#V1" title="FRECHET:attr.1">first-countable</a> implies ( <font color="Maroon" title="c1">T</font> <a href="compts_1.html#NR1" title="COMPTS_1:NR.1">is_T2</a>  iff for <font color="Olive" title="b1">S</font> being  <a href="struct_0.html#NM9" title="STRUCT_0:NM.9">sequence</a> of <font color="Maroon" title="c1">T</font><br/> for <font color="Olive" title="b2">x1</font>, <font color="Olive" title="b3">x2</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">T</font>  st <font color="Olive" title="b2">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="frechet.html#K1" title="FRECHET:func.1">Lim</a> <font color="Olive" title="b1">S</font> &amp; <font color="Olive" title="b3">x2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="frechet.html#K1" title="FRECHET:func.1">Lim</a> <font color="Olive" title="b1">S</font> holds <br/><font color="Olive" title="b2">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">x2</font> ) )</span><br/>

<b>assume </b><a NAME="E1:22"/><i><font color="Green" title="E18">A1</font></i>: 
<font color="Maroon" title="c1">T</font> is <a href="frechet.html#V1" title="FRECHET:attr.1">first-countable</a>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c1">T</font> <a href="compts_1.html#NR1" title="COMPTS_1:NR.1">is_T2</a>  iff for <font color="Olive" title="b1">S</font> being  <a href="struct_0.html#NM9" title="STRUCT_0:NM.9">sequence</a> of <font color="Maroon" title="c1">T</font><br/> for <font color="Olive" title="b2">x1</font>, <font color="Olive" title="b3">x2</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">T</font>  st <font color="Olive" title="b2">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="frechet.html#K1" title="FRECHET:func.1">Lim</a> <font color="Olive" title="b1">S</font> &amp; <font color="Olive" title="b3">x2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="frechet.html#K1" title="FRECHET:func.1">Lim</a> <font color="Olive" title="b1">S</font> holds <br/><font color="Olive" title="b2">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">x2</font> )</span><br/>

<b>thus </b><a NAME="E2:22"/>
( <font color="Maroon" title="c1">T</font> <a href="compts_1.html#NR1" title="COMPTS_1:NR.1">is_T2</a>  implies for <font color="Olive" title="b1">S</font> being  <a href="struct_0.html#NM9" title="STRUCT_0:NM.9">sequence</a> of <font color="Maroon" title="c1">T</font><br/> for <font color="Olive" title="b2">x1</font>, <font color="Olive" title="b3">x2</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">T</font>  st <font color="Olive" title="b2">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="frechet.html#K1" title="FRECHET:func.1">Lim</a> <font color="Olive" title="b1">S</font> &amp; <font color="Olive" title="b3">x2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="frechet.html#K1" title="FRECHET:func.1">Lim</a> <font color="Olive" title="b1">S</font> holds <br/><font color="Olive" title="b2">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">x2</font> )
 <b>by </b><i><a class="ref" href="frechet2.html#T9" target="_self" title="FRECHET2:th.9">Th9</a></i>; <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">S</font> being  <a href="struct_0.html#NM9" title="STRUCT_0:NM.9">sequence</a> of <font color="Maroon" title="c1">T</font><br/> for <font color="Olive" title="b2">x1</font>, <font color="Olive" title="b3">x2</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">T</font>  st <font color="Olive" title="b2">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="frechet.html#K1" title="FRECHET:func.1">Lim</a> <font color="Olive" title="b1">S</font> &amp; <font color="Olive" title="b3">x2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="frechet.html#K1" title="FRECHET:func.1">Lim</a> <font color="Olive" title="b1">S</font> holds <br/><font color="Olive" title="b2">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">x2</font> ) implies <font color="Maroon" title="c1">T</font> <a href="compts_1.html#NR1" title="COMPTS_1:NR.1">is_T2</a>  )</span><br/>

<b>assume </b><a NAME="E3:22"/><i><font color="Green" title="E19">A2</font></i>: 
for <font color="Olive" title="b1">S</font> being  <a href="struct_0.html#NM9" title="STRUCT_0:NM.9">sequence</a> of <font color="Maroon" title="c1">T</font><br/> for <font color="Olive" title="b2">x1</font>, <font color="Olive" title="b3">x2</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">T</font>  st <font color="Olive" title="b2">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="frechet.html#K1" title="FRECHET:func.1">Lim</a> <font color="Olive" title="b1">S</font> &amp; <font color="Olive" title="b3">x2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="frechet.html#K1" title="FRECHET:func.1">Lim</a> <font color="Olive" title="b1">S</font> holds <br/><font color="Olive" title="b2">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">x2</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c1">T</font> <a href="compts_1.html#NR1" title="COMPTS_1:NR.1">is_T2</a> </span><br/>

<b>assume </b><a NAME="E4:22"/>
not <font color="Maroon" title="c1">T</font> <a href="compts_1.html#NR1" title="COMPTS_1:NR.1">is_T2</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/>

<b>then consider </b><font color="Maroon" title="c2">x1</font>, <font color="Maroon" title="c3">x2</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">T</font><b> such that </b><br/><a NAME="E6:22"/><i><font color="Green" title="E20">A3</font></i>: 
<font color="Maroon" title="c2">x1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">x2</font>
 <b>and </b><br/><a NAME="E7:22"/><i><font color="Green" title="E21">A4</font></i>: 
for <font color="Olive" title="b1">U1</font>, <font color="Olive" title="b2">U2</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">T</font>  st <font color="Olive" title="b1">U1</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> &amp; <font color="Olive" title="b2">U2</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> &amp; <font color="Maroon" title="c2">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">U1</font> &amp; <font color="Maroon" title="c3">x2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b2">U2</font> holds <br/><font color="Olive" title="b1">U1</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Olive" title="b2">U2</font>
 <b>by </b><i><a class="ref" href="compts_1.html#D4" title="COMPTS_1:def.4">COMPTS_1:def 4</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c4">B1</font> being    <a href="yellow_8.html#M1" title="YELLOW_8:mode.1">Basis</a> of <font color="Maroon" title="c2">x1</font><b> such that </b><br/><a NAME="E9:22"/><i><font color="Green" title="E22">A5</font></i>: 
 ex <font color="Olive" title="b1">S</font> being  <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a> st <br/>(  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Olive" title="b1">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  &amp;  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Olive" title="b1">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">B1</font> &amp; ( for <font color="Olive" title="b2">n</font>, <font color="Olive" title="b3">m</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>   st <font color="Olive" title="b3">m</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Olive" title="b2">n</font> holds <br/><font color="Olive" title="b1">S</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b3">m</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">S</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">n</font> ) )
 <b>by </b><i><a class="txt" href="frechet2.html#E1:22"><i><font color="Green" title="E18">A1</font></i></a>, <a class="txt" href="frechet2.html#E17"><i><font color="Green" title="E16">Lm10</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon" title="c5">S1</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><b> such that </b><br/><a NAME="E11:22"/><i><font color="Green" title="E23">A6</font></i>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c5">S1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 <b>and </b><br/><a NAME="E12:22"/><i><font color="Green" title="E24">A7</font></i>: 
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c5">S1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">B1</font>
 <b>and </b><br/><a NAME="E13:22"/><i><font color="Green" title="E25">A8</font></i>: 
for <font color="Olive" title="b1">n</font>, <font color="Olive" title="b2">m</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>   st <font color="Olive" title="b2">m</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Olive" title="b1">n</font> holds <br/><font color="Maroon" title="c5">S1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">m</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">S1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">n</font>
 <b>by </b><i><a class="txt" href="frechet2.html#E9:22"><i><font color="Green" title="E22">A5</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon" title="c6">B2</font> being    <a href="yellow_8.html#M1" title="YELLOW_8:mode.1">Basis</a> of <font color="Maroon" title="c3">x2</font><b> such that </b><br/><a NAME="E15:22"/><i><font color="Green" title="E26">A9</font></i>: 
 ex <font color="Olive" title="b1">S</font> being  <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a> st <br/>(  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Olive" title="b1">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  &amp;  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Olive" title="b1">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">B2</font> &amp; ( for <font color="Olive" title="b2">n</font>, <font color="Olive" title="b3">m</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>   st <font color="Olive" title="b3">m</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Olive" title="b2">n</font> holds <br/><font color="Olive" title="b1">S</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b3">m</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">S</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">n</font> ) )
 <b>by </b><i><a class="txt" href="frechet2.html#E1:22"><i><font color="Green" title="E18">A1</font></i></a>, <a class="txt" href="frechet2.html#E17"><i><font color="Green" title="E16">Lm10</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon" title="c7">S2</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><b> such that </b><br/><a NAME="E17:22"/><i><font color="Green" title="E27">A10</font></i>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c7">S2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 <b>and </b><br/><a NAME="E18:22"/><i><font color="Green" title="E28">A11</font></i>: 
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c7">S2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">B2</font>
 <b>and </b><br/><a NAME="E19:22"/><i><font color="Green" title="E29">A12</font></i>: 
for <font color="Olive" title="b1">n</font>, <font color="Olive" title="b2">m</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>   st <font color="Olive" title="b2">m</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Olive" title="b1">n</font> holds <br/><font color="Maroon" title="c7">S2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">m</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c7">S2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">n</font>
 <b>by </b><i><a class="txt" href="frechet2.html#E15:22"><i><font color="Green" title="E26">A9</font></i></a></i>;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ,   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <b>means </b>$2 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">S1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> $1</span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">S2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> $1</span>)</span>;<br/>
<a NAME="E20:22"/><i><font color="Green" title="E30">A13</font></i>: 
for <font color="Olive" title="b1">n</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds <br/> ex <font color="Olive" title="b2">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">n</font>,<font color="Olive" title="b2">x</font>] )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c8">n</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">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  implies  ex <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c8">n</font>,<font color="Olive" title="b1">x</font>] ) )</span><br/>

<b>assume </b><a NAME="E1:22_1"/><i><font color="Green" title="E31">A14</font></i>: 
<font color="Maroon" title="c8">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c8">n</font>,<font color="Olive" title="b1">x</font>] )</span><br/>

<a NAME="E2:22_1"/><b>then </b><i><font color="Green" title="E32">A15</font></i>: 
<font color="Maroon" title="c5">S1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">B1</font>
 
<b>by </b><i><a class="txt" href="frechet2.html#E11:22"><i><font color="Green" title="E23">A6</font></i></a>, <a class="txt" href="frechet2.html#E12:22"><i><font color="Green" title="E24">A7</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c9">U1</font> = <font color="Maroon" title="c5">S1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">n</font> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">T</font> ;<br/>
<b>reconsider </b><font color="Maroon" title="c10">U1</font> = <font color="Maroon" title="c9">U1</font> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">T</font> ;<br/>
<a NAME="E5:22_1"/><i><font color="Green" title="E33">A16</font></i>: 
( <font color="Maroon" title="c10">U1</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> &amp; <font color="Maroon" title="c2">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">U1</font> )
 
<b>by </b><i><a class="txt" href="frechet2.html#E2:22_1"><i><font color="Green" title="E32">A15</font></i></a>, <a class="ref" href="yellow_8.html#T21" title="YELLOW_8:th.21">YELLOW_8:21</a></i>;<br/>
<a NAME="E6:22_1"/><i><font color="Green" title="E34">A17</font></i>: 
<font color="Maroon" title="c7">S2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">B2</font>
 
<b>by </b><i><a class="txt" href="frechet2.html#E17:22"><i><font color="Green" title="E27">A10</font></i></a>, <a class="txt" href="frechet2.html#E18:22"><i><font color="Green" title="E28">A11</font></i></a>, <a class="txt" href="frechet2.html#E1:22_1"><i><font color="Green" title="E31">A14</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c11">U2</font> = <font color="Maroon" title="c7">S2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">n</font> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">T</font> ;<br/>
<b>reconsider </b><font color="Maroon" title="c12">U2</font> = <font color="Maroon" title="c11">U2</font> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">T</font> ;<br/>
<a NAME="E9:22_1"/>
( <font color="Maroon" title="c12">U2</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> &amp; <font color="Maroon" title="c3">x2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c12">U2</font> )
 
<b>by </b><i><a class="txt" href="frechet2.html#E6:22_1"><i><font color="Green" title="E34">A17</font></i></a>, <a class="ref" href="yellow_8.html#T21" title="YELLOW_8:th.21">YELLOW_8:21</a></i>;<br/>
<a NAME="E10:22_1"/><b>then </b>
<font color="Maroon" title="c10">U1</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c12">U2</font>
 
<b>by </b><i><a class="txt" href="frechet2.html#E7:22"><i><font color="Green" title="E21">A4</font></i></a>, <a class="txt" href="frechet2.html#E5:22_1"><i><font color="Green" title="E33">A16</font></i></a></i>;<br/>
<a NAME="E11:22_1"/><b>then </b><i><font color="Green" title="E35">A18</font></i>: 
<font color="Maroon" title="c10">U1</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c12">U2</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 
<b>by </b><i><a class="ref" href="xboole_0.html#D7" title="XBOOLE_0:def.7">XBOOLE_0:def 7</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c13">x</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c5">S1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">n</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">S2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">n</font></span>)</span>;<br/>
<b>take </b>
<font color="Maroon" title="c13">x</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">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c8">n</font>,<font color="Maroon" title="c13">x</font>] )</span><br/>

<a NAME="E13:22_1"/>
<font color="Maroon" title="c13">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">U1</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c12">U2</font>
 
<b>by </b><i><a class="txt" href="frechet2.html#E11:22_1"><i><font color="Green" title="E35">A18</font></i></a></i>;<br/>
<b>hence </b><a NAME="E14:22_1"/>
<font color="Maroon" title="c13">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c8">n</font>,<font color="Maroon" title="c13">x</font>]</span><br/>

<b>thus </b><a NAME="E15:22_1"/>
<font color="Maroon" title="c13">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">S1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">n</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">S2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">n</font></span>)</span>
 <b>by </b><i><a class="txt" href="frechet2.html#E11:22_1"><i><font color="Green" title="E35">A18</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>
<b>consider </b><font color="Maroon" title="c8">S</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><b> such that </b><br/><a NAME="E22:22"/><i><font color="Green" title="E31">A19</font></i>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c8">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 <b>and </b><br/><a NAME="E23:22"/><i><font color="Green" title="E32">A20</font></i>: 
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c8">S</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font>
 <b>and </b><br/><a NAME="E24:22"/><i><font color="Green" title="E33">A21</font></i>: 
for <font color="Olive" title="b1">n</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">n</font>,<font color="Maroon" title="c8">S</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">n</font>]
 <b>from </b><i><a class="ref" href="wellord2.html#S1" title="WELLORD2:sch.1">WELLORD2:sch 1</a>(<a class="txt" href="frechet2.html#E20:22"><i><font color="Green" title="E30">A13</font></i></a>);<br/></i>
<b>reconsider </b><font color="Maroon" title="c9">S</font> = <font color="Maroon" title="c8">S</font> as   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font> <b>by </b><i><a class="txt" href="frechet2.html#E22:22"><i><font color="Green" title="E31">A19</font></i></a>, <a class="txt" href="frechet2.html#E23:22"><i><font color="Green" title="E32">A20</font></i></a>, <a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a>, <a class="ref" href="relset_1.html#T11" title="RELSET_1:th.11">RELSET_1:11</a></i>;<br/>
<a NAME="E26:22"/>
<font color="Maroon" title="c9">S</font> <a href="frechet.html#R1" title="FRECHET:pred.1">is_convergent_to</a> <font color="Maroon" title="c2">x1</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c10">U1</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">T</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="frechet.html#D2" title="FRECHET:def.2">FRECHET:def 2</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c10">U1</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> or  not <font color="Maroon" title="c2">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">U1</font> or  ex <font color="Olive">b<sub>1</sub></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>  st <br/>for <font color="Olive">b<sub>2</sub></font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   holds <br/>(  not <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive">b<sub>2</sub></font> or <font color="Maroon" title="c9">S</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">U1</font> ) )</span><br/>

<b>assume </b><a NAME="E1:22_2"/>
( <font color="Maroon" title="c10">U1</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> &amp; <font color="Maroon" title="c2">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">U1</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive">b<sub>1</sub></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>  st <br/>for <font color="Olive">b<sub>2</sub></font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   holds <br/>(  not <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive">b<sub>2</sub></font> or <font color="Maroon" title="c9">S</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">U1</font> )</span><br/>

<b>then consider </b><font color="Maroon" title="c11">V1</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">T</font><b> such that </b><br/><a NAME="E3:22_2"/><i><font color="Green" title="E34">A22</font></i>: 
<font color="Maroon" title="c11">V1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">B1</font>
 <b>and </b><br/><a NAME="E4:22_2"/><i><font color="Green" title="E35">A23</font></i>: 
<font color="Maroon" title="c11">V1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c10">U1</font>
 <b>by </b><i><a class="ref" href="yellow_8.html#T22" title="YELLOW_8:th.22">YELLOW_8:22</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c12">n</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E6:22_2"/><i><font color="Green" title="E36">A24</font></i>: 
<font color="Maroon" title="c12">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c5">S1</font>
 <b>and </b><br/><a NAME="E7:22_2"/><i><font color="Green" title="E37">A25</font></i>: 
<font color="Maroon" title="c11">V1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">S1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">n</font>
 <b>by </b><i><a class="txt" href="frechet2.html#E12:22"><i><font color="Green" title="E24">A7</font></i></a>, <a class="txt" href="frechet2.html#E3:22_2"><i><font color="Green" title="E34">A22</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c13">n</font> = <font color="Maroon" title="c12">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="frechet2.html#E11:22"><i><font color="Green" title="E23">A6</font></i></a>, <a class="txt" href="frechet2.html#E6:22_2"><i><font color="Green" title="E36">A24</font></i></a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c13">n</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   holds <br/>(  not <font color="Maroon" title="c13">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> or <font color="Maroon" title="c9">S</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">U1</font> )</span><br/>

<b>let </b><font color="Maroon" title="c14">m</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c13">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c14">m</font> or <font color="Maroon" title="c9">S</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c14">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">U1</font> )</span><br/>

<b>assume </b><a NAME="E9:22_2"/>
<font color="Maroon" title="c13">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c14">m</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c9">S</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c14">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">U1</font></span><br/>

<a NAME="E10:22_2"/><b>then </b><i><font color="Green" title="E38">A26</font></i>: 
<font color="Maroon" title="c5">S1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">m</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">S1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">n</font>
 
<b>by </b><i><a class="txt" href="frechet2.html#E13:22"><i><font color="Green" title="E25">A8</font></i></a></i>;<br/>
<a NAME="E11:22_2"/><i><font color="Green" title="E39">A27</font></i>: 
<font color="Maroon" title="c9">S</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c14">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">S1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">m</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">S2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">m</font></span>)</span>
 
<b>by </b><i><a class="txt" href="frechet2.html#E24:22"><i><font color="Green" title="E33">A21</font></i></a></i>;<br/>
<a NAME="E12:22_2"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c5">S1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">m</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">S2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">m</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">S1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">m</font>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T17" title="XBOOLE_1:th.17">XBOOLE_1:17</a></i>;<br/>
<a NAME="E13:22_2"/><b>then </b>
<font color="Maroon" title="c9">S</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c14">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">S1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">m</font>
 
<b>by </b><i><a class="txt" href="frechet2.html#E11:22_2"><i><font color="Green" title="E39">A27</font></i></a></i>;<br/>
<a NAME="E14:22_2"/><b>then </b>
<font color="Maroon" title="c9">S</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c14">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">S1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">n</font>
 
<b>by </b><i><a class="txt" href="frechet2.html#E10:22_2"><i><font color="Green" title="E38">A26</font></i></a></i>;<br/>
<b>hence </b><a NAME="E15:22_2"/>
<font color="Maroon" title="c9">S</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c14">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">U1</font>
 <b>by </b><i><a class="txt" href="frechet2.html#E4:22_2"><i><font color="Green" title="E35">A23</font></i></a>, <a class="txt" href="frechet2.html#E7:22_2"><i><font color="Green" title="E37">A25</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="E27:22"/><b>then </b><i><font color="Green" title="E34">A28</font></i>: 
<font color="Maroon" title="c2">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="frechet.html#K1" title="FRECHET:func.1">Lim</a> <font color="Maroon" title="c9">S</font>
 
<b>by </b><i><a class="ref" href="frechet.html#D4" title="FRECHET:def.4">FRECHET:def 4</a></i>;<br/>
<a NAME="E28:22"/>
<font color="Maroon" title="c9">S</font> <a href="frechet.html#R1" title="FRECHET:pred.1">is_convergent_to</a> <font color="Maroon" title="c3">x2</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c10">U2</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">T</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="frechet.html#D2" title="FRECHET:def.2">FRECHET:def 2</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c10">U2</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> or  not <font color="Maroon" title="c3">x2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">U2</font> or  ex <font color="Olive">b<sub>1</sub></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>  st <br/>for <font color="Olive">b<sub>2</sub></font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   holds <br/>(  not <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive">b<sub>2</sub></font> or <font color="Maroon" title="c9">S</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">U2</font> ) )</span><br/>

<b>assume </b><a NAME="E1:22_3"/>
( <font color="Maroon" title="c10">U2</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> &amp; <font color="Maroon" title="c3">x2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">U2</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive">b<sub>1</sub></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>  st <br/>for <font color="Olive">b<sub>2</sub></font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   holds <br/>(  not <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive">b<sub>2</sub></font> or <font color="Maroon" title="c9">S</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">U2</font> )</span><br/>

<b>then consider </b><font color="Maroon" title="c11">V2</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">T</font><b> such that </b><br/><a NAME="E3:22_3"/><i><font color="Green" title="E35">A29</font></i>: 
<font color="Maroon" title="c11">V2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">B2</font>
 <b>and </b><br/><a NAME="E4:22_3"/><i><font color="Green" title="E36">A30</font></i>: 
<font color="Maroon" title="c11">V2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c10">U2</font>
 <b>by </b><i><a class="ref" href="yellow_8.html#T22" title="YELLOW_8:th.22">YELLOW_8:22</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c12">n</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E6:22_3"/><i><font color="Green" title="E37">A31</font></i>: 
<font color="Maroon" title="c12">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c7">S2</font>
 <b>and </b><br/><a NAME="E7:22_3"/><i><font color="Green" title="E38">A32</font></i>: 
<font color="Maroon" title="c11">V2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">S2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">n</font>
 <b>by </b><i><a class="txt" href="frechet2.html#E18:22"><i><font color="Green" title="E28">A11</font></i></a>, <a class="txt" href="frechet2.html#E3:22_3"><i><font color="Green" title="E35">A29</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c13">n</font> = <font color="Maroon" title="c12">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="frechet2.html#E17:22"><i><font color="Green" title="E27">A10</font></i></a>, <a class="txt" href="frechet2.html#E6:22_3"><i><font color="Green" title="E37">A31</font></i></a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c13">n</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   holds <br/>(  not <font color="Maroon" title="c13">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> or <font color="Maroon" title="c9">S</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">U2</font> )</span><br/>

<b>let </b><font color="Maroon" title="c14">m</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c13">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c14">m</font> or <font color="Maroon" title="c9">S</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c14">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">U2</font> )</span><br/>

<b>assume </b><a NAME="E9:22_3"/>
<font color="Maroon" title="c13">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c14">m</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c9">S</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c14">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">U2</font></span><br/>

<a NAME="E10:22_3"/><b>then </b><i><font color="Green" title="E39">A33</font></i>: 
<font color="Maroon" title="c7">S2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">m</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c7">S2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">n</font>
 
<b>by </b><i><a class="txt" href="frechet2.html#E19:22"><i><font color="Green" title="E29">A12</font></i></a></i>;<br/>
<a NAME="E11:22_3"/><i><font color="Green" title="E40">A34</font></i>: 
<font color="Maroon" title="c9">S</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c14">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">S1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">m</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">S2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">m</font></span>)</span>
 
<b>by </b><i><a class="txt" href="frechet2.html#E24:22"><i><font color="Green" title="E33">A21</font></i></a></i>;<br/>
<a NAME="E12:22_3"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c5">S1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">m</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">S2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">m</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c7">S2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">m</font>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T17" title="XBOOLE_1:th.17">XBOOLE_1:17</a></i>;<br/>
<a NAME="E13:22_3"/><b>then </b>
<font color="Maroon" title="c9">S</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c14">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">S2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">m</font>
 
<b>by </b><i><a class="txt" href="frechet2.html#E11:22_3"><i><font color="Green" title="E40">A34</font></i></a></i>;<br/>
<a NAME="E14:22_3"/><b>then </b>
<font color="Maroon" title="c9">S</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c14">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">S2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">n</font>
 
<b>by </b><i><a class="txt" href="frechet2.html#E10:22_3"><i><font color="Green" title="E39">A33</font></i></a></i>;<br/>
<b>hence </b><a NAME="E15:22_3"/>
<font color="Maroon" title="c9">S</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c14">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">U2</font>
 <b>by </b><i><a class="txt" href="frechet2.html#E4:22_3"><i><font color="Green" title="E36">A30</font></i></a>, <a class="txt" href="frechet2.html#E7:22_3"><i><font color="Green" title="E38">A32</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="E29:22"/><b>then </b>
<font color="Maroon" title="c3">x2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="frechet.html#K1" title="FRECHET:func.1">Lim</a> <font color="Maroon" title="c9">S</font>
 
<b>by </b><i><a class="ref" href="frechet.html#D4" title="FRECHET:def.4">FRECHET:def 4</a></i>;<br/>
<b>hence </b><a NAME="E30:22"/>
contradiction
 <b>by </b><i><a class="txt" href="frechet2.html#E3:22"><i><font color="Green" title="E19">A2</font></i></a>, <a class="txt" href="frechet2.html#E6:22"><i><font color="Green" title="E20">A3</font></i></a>, <a class="txt" href="frechet2.html#E27:22"><i><font color="Green" title="E34">A28</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>
