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<span class="kw">let </span><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:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c1">T</font> is  <a href="pre_topc.html#V8" title="PRE_TOPC:attr.8">T_2</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#K3" title="FRECHET:func.3">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#K3" title="FRECHET:func.3">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/>

<span class="kw">assume </span><a NAME="E1:12"/><span class="lab"><font color="Green" title="E9">A1</font></span>: 
<font color="Maroon" title="c1">T</font> is  <a href="pre_topc.html#V8" title="PRE_TOPC:attr.8">T_2</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">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#K3" title="FRECHET:func.3">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#K3" title="FRECHET:func.3">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/>

<span class="kw">assume </span><a NAME="E2:12"/>
 ex <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> ex <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 <br/>( <font color="Olive" title="b2">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="frechet.html#K3" title="FRECHET:func.3">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#K3" title="FRECHET:func.3">Lim</a> <font color="Olive" title="b1">S</font> &amp; not <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:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c2">S</font> being   <a href="struct_0.html#NM9" title="STRUCT_0:NM.9">sequence</a> of <font color="Maroon" title="c1">T</font><span class="kw"> such that </span><br/><a NAME="E4:12"/><span class="lab"><font color="Green" title="E10">A2</font></span>: 
 ex <font color="Olive" title="b1">x1</font>, <font color="Olive" title="b2">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 <br/>( <font color="Olive" title="b1">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="frechet.html#K3" title="FRECHET:func.3">Lim</a> <font color="Maroon" title="c2">S</font> &amp; <font color="Olive" title="b2">x2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="frechet.html#K3" title="FRECHET:func.3">Lim</a> <font color="Maroon" title="c2">S</font> &amp; <font color="Olive" title="b1">x1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b2">x2</font> )
 ;<br/>
<span class="kw">consider </span><font color="Maroon" title="c3">x1</font>, <font color="Maroon" title="c4">x2</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">T</font><span class="kw"> such that </span><br/><a NAME="E6:12"/><span class="lab"><font color="Green" title="E11">A3</font></span>: 
<font color="Maroon" title="c3">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="frechet.html#K3" title="FRECHET:func.3">Lim</a> <font color="Maroon" title="c2">S</font>
 <span class="kw">and </span><br/><a NAME="E7:12"/><span class="lab"><font color="Green" title="E12">A4</font></span>: 
<font color="Maroon" title="c4">x2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="frechet.html#K3" title="FRECHET:func.3">Lim</a> <font color="Maroon" title="c2">S</font>
 <span class="kw">and </span><br/><a NAME="E8:12"/><span class="lab"><font color="Green" title="E13">A5</font></span>: 
<font color="Maroon" title="c3">x1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">x2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:12"><span class="lab"><font color="Green" title="E10">A2</font></span></a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c5">U1</font>, <font color="Maroon" title="c6">U2</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">T</font><span class="kw"> such that </span><br/><a NAME="E10:12"/><span class="lab"><font color="Green" title="E14">A6</font></span>: 
<font color="Maroon" title="c5">U1</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> 
 <span class="kw">and </span><br/><a NAME="E11:12"/><span class="lab"><font color="Green" title="E15">A7</font></span>: 
<font color="Maroon" title="c6">U2</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> 
 <span class="kw">and </span><br/><a NAME="E12:12"/><span class="lab"><font color="Green" title="E16">A8</font></span>: 
<font color="Maroon" title="c3">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">U1</font>
 <span class="kw">and </span><br/><a NAME="E13:12"/><span class="lab"><font color="Green" title="E17">A9</font></span>: 
<font color="Maroon" title="c4">x2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">U2</font>
 <span class="kw">and </span><br/><a NAME="E14:12"/><span class="lab"><font color="Green" title="E18">A10</font></span>: 
<font color="Maroon" title="c5">U1</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c6">U2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:12"><span class="lab"><font color="Green" title="E9">A1</font></span></a>, <a class="txt" href="#E8:12"><span class="lab"><font color="Green" title="E13">A5</font></span></a>, <a class="ref" href="pre_topc.html#D16" title="PRE_TOPC:def.16">PRE_TOPC:def 16</a></span>;<br/>
<a NAME="E15:12"/>
<font color="Maroon" title="c2">S</font> <a href="frechet.html#R1" title="FRECHET:pred.1">is_convergent_to</a> <font color="Maroon" title="c3">x1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:12"><span class="lab"><font color="Green" title="E11">A3</font></span></a>, <a class="ref" href="frechet.html#D5" title="FRECHET:def.5">FRECHET:def 5</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c7">n1</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <span class="kw"> such that </span><br/><a NAME="E17:12"/><span class="lab"><font color="Green" title="E19">A11</font></span>: 
for <font color="Olive" title="b1">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="Maroon" title="c7">n1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">m</font> holds <br/><font color="Maroon" title="c2">S</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Olive" title="b1">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">U1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:12"><span class="lab"><font color="Green" title="E14">A6</font></span></a>, <a class="txt" href="#E12:12"><span class="lab"><font color="Green" title="E16">A8</font></span></a>, <a class="ref" href="frechet.html#D3" title="FRECHET:def.3">FRECHET:def 3</a></span>;<br/>
<a NAME="E18:12"/>
<font color="Maroon" title="c2">S</font> <a href="frechet.html#R1" title="FRECHET:pred.1">is_convergent_to</a> <font color="Maroon" title="c4">x2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:12"><span class="lab"><font color="Green" title="E12">A4</font></span></a>, <a class="ref" href="frechet.html#D5" title="FRECHET:def.5">FRECHET:def 5</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c8">n2</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <span class="kw"> such that </span><br/><a NAME="E20:12"/><span class="lab"><font color="Green" title="E20">A12</font></span>: 
for <font color="Olive" title="b1">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="Maroon" title="c8">n2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">m</font> holds <br/><font color="Maroon" title="c2">S</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Olive" title="b1">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">U2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:12"><span class="lab"><font color="Green" title="E15">A7</font></span></a>, <a class="txt" href="#E13:12"><span class="lab"><font color="Green" title="E17">A9</font></span></a>, <a class="ref" href="frechet.html#D3" title="FRECHET:def.3">FRECHET:def 3</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c9">n</font> =  <a href="limfunc1.html#K1" title="LIMFUNC1:func.1">max</a> <font color="Maroon" title="c7">n1</font>,<font color="Maroon" title="c8">n2</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  ;<br/>
<a NAME="E22:12"/><span class="lab"><font color="Green" title="E21">A13</font></span>: 
<font color="Maroon" title="c2">S</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c9">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">U1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:12"><span class="lab"><font color="Green" title="E19">A11</font></span></a>, <a class="ref" href="xxreal_0.html#T25" title="XXREAL_0:th.25">XXREAL_0:25</a></span>;<br/>
<a NAME="E23:12"/><span class="lab"><font color="Green" title="E22">A14</font></span>: 
<font color="Maroon" title="c2">S</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c9">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">U2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:12"><span class="lab"><font color="Green" title="E20">A12</font></span></a>, <a class="ref" href="xxreal_0.html#T25" title="XXREAL_0:th.25">XXREAL_0:25</a></span>;<br/>
<a NAME="E24:12"/>
<font color="Maroon" title="c5">U1</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c6">U2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:12"><span class="lab"><font color="Green" title="E18">A10</font></span></a>, <a class="ref" href="xboole_0.html#D7" title="XBOOLE_0:def.7">XBOOLE_0:def 7</a></span>;<br/>
<span class="kw">hence </span><a NAME="E25:12"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E22:12"><span class="lab"><font color="Green" title="E21">A13</font></span></a>, <a class="txt" href="#E23:12"><span class="lab"><font color="Green" title="E22">A14</font></span></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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