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<div class="add">

<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="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">r</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a><br/>  for <font color="Olive" title="b2">f</font> being   <a href="pscomp_1.html#NM1" title="PSCOMP_1:NM.1">RealMap</a> of <font color="Maroon" title="c1">T</font>  st <font color="Olive" title="b2">f</font> is  <a href="pscomp_1.html#V1" title="PSCOMP_1:attr.1">continuous</a>  holds <br/> <a href="nagata_1.html#K5" title="NAGATA_1:func.5">min</a> (<font color="Olive" title="b1">r</font>,<font color="Olive" title="b2">f</font>) is  <a href="pscomp_1.html#V1" title="PSCOMP_1:attr.1">continuous</a> </span><br/><span class="kw">let </span><font color="Maroon" title="c2">r</font> be   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">f</font> being   <a href="pscomp_1.html#NM1" title="PSCOMP_1:NM.1">RealMap</a> of <font color="Maroon" title="c1">T</font>  st <font color="Olive" title="b1">f</font> is  <a href="pscomp_1.html#V1" title="PSCOMP_1:attr.1">continuous</a>  holds <br/> <a href="nagata_1.html#K5" title="NAGATA_1:func.5">min</a> (<font color="Maroon" title="c2">r</font>,<font color="Olive" title="b1">f</font>) is  <a href="pscomp_1.html#V1" title="PSCOMP_1:attr.1">continuous</a> </span><br/><span class="kw">let </span><font color="Maroon" title="c3">f</font> be   <a href="pscomp_1.html#NM1" title="PSCOMP_1:NM.1">RealMap</a> of <font color="Maroon" title="c1">T</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c3">f</font> is  <a href="pscomp_1.html#V1" title="PSCOMP_1:attr.1">continuous</a>  implies  <a href="nagata_1.html#K5" title="NAGATA_1:func.5">min</a> (<font color="Maroon" title="c2">r</font>,<font color="Maroon" title="c3">f</font>) is  <a href="pscomp_1.html#V1" title="PSCOMP_1:attr.1">continuous</a>  )</span><br/>





<span class="kw">assume </span><a NAME="E1:51"/><span class="lab"><font color="Green" title="E28">A1</font></span>: 
<font color="Maroon" title="c3">f</font> is  <a href="pscomp_1.html#V1" title="PSCOMP_1:attr.1">continuous</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="nagata_1.html#K5" title="NAGATA_1:func.5">min</a> (<font color="Maroon" title="c2">r</font>,<font color="Maroon" title="c3">f</font>) is  <a href="pscomp_1.html#V1" title="PSCOMP_1:attr.1">continuous</a> </span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c4">f9</font> = <font color="Maroon" title="c3">f</font>, <font color="Maroon" title="c5">mf9</font> =  <a href="nagata_1.html#K5" title="NAGATA_1:func.5">min</a> (<font color="Maroon" title="c2">r</font>,<font color="Maroon" title="c3">f</font>) as   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c1">T</font>,<a href="topmetr.html#K3" title="TOPMETR:func.3">R^1</a> <span class="kw">by</span> <span class="lab"><a class="ref" href="topmetr.html#T17" title="TOPMETR:th.17">TOPMETR:17</a></span>;<br/>
<a NAME="E3:51"/><span class="lab"><font color="Green" title="E29">A2</font></span>: 
<font color="Maroon" title="c4">f9</font> is  <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:51"><span class="lab"><font color="Green" title="E28">A1</font></span></a>, <a class="ref" href="topreal6.html#T74" title="TOPREAL6:th.74">TOPREAL6:74</a></span>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="51_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c6">t</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">T</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c5">mf9</font> <a href="tmap_1.html#R1" title="TMAP_1:pred.1">is_continuous_at</a> <font color="Maroon" title="c6">t</font></span><br/><a NAME="E1:51_1"/>
 for <font color="Olive" title="b1">R</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <a href="topmetr.html#K3" title="TOPMETR:func.3">R^1</a>  st <font color="Olive" title="b1">R</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  &amp; <font color="Maroon" title="c5">mf9</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">R</font> holds <br/> ex <font color="Olive" title="b2">U</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">T</font> st <br/>( <font color="Olive" title="b2">U</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  &amp; <font color="Maroon" title="c6">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b2">U</font> &amp; <font color="Maroon" title="c5">mf9</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Olive" title="b2">U</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">R</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="51_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">reconsider </span><font color="Maroon" title="c7">ft</font> = <font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">t</font> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <a href="metric_1.html#K8" title="METRIC_1:func.8">RealSpace</a> <span class="kw">by</span> <span class="lab"><a class="ref" href="metric_1.html#D13" title="METRIC_1:def.13">METRIC_1:def 13</a></span>;<br/>
<span class="kw">let </span><font color="Maroon" title="c8">R</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <a href="topmetr.html#K3" title="TOPMETR:func.3">R^1</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c8">R</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  &amp; <font color="Maroon" title="c5">mf9</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">R</font> implies  ex <font color="Olive" title="b1">U</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">T</font> st <br/>( <font color="Olive" title="b1">U</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  &amp; <font color="Maroon" title="c6">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">U</font> &amp; <font color="Maroon" title="c5">mf9</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Olive" title="b1">U</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c8">R</font> ) )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E2:51_1_1"/><span class="lab"><font color="Green" title="E30">A3</font></span>: 
<font color="Maroon" title="c8">R</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> 
 <span class="kw">and </span><br/><a NAME="E3:51_1_1"/><span class="lab"><font color="Green" title="E31">A4</font></span>: 
<font color="Maroon" title="c5">mf9</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">R</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">U</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">T</font> st <br/>( <font color="Olive" title="b1">U</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  &amp; <font color="Maroon" title="c6">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">U</font> &amp; <font color="Maroon" title="c5">mf9</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Olive" title="b1">U</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c8">R</font> )</span><br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="51_1_1_1"><span class="kw">now </span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="51_1_1_1_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:51_1_1_1_1"/>
( <font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">t</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">r</font> or <font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">t</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c2">r</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="51_1_1_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:51_1_1_1_1_1"/><span class="lab"><font color="Green" title="E32">A5</font></span>: 
<font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">t</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">r</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">U</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">T</font> st <br/>( <font color="Olive" title="b1">U</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  &amp; <font color="Maroon" title="c6">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">U</font> &amp; <span class="p1">(<span class="default"><a href="nagata_1.html#K5" title="NAGATA_1:func.5">min</a> (<font color="Maroon" title="c2">r</font>,<font color="Maroon" title="c3">f</font>)</span>)</span> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Olive" title="b1">U</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c8">R</font> )</span><br/><div class="add"><a NAME="E2:51_1_1_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xxreal_0.html#K3" title="XXREAL_0:func.3">min</a> (<font color="Maroon" title="c2">r</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">t</font></span>)</span>)
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_0.html#D9" title="XXREAL_0:def.9">XXREAL_0:def 9</a></span>;<br/><a NAME="E3:51_1_1_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c7">ft</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">R</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:51_1_1"><span class="lab"><font color="Green" title="E31">A4</font></span></a>, <a class="ref" href="nagata_1.html#D9" target="_self" title="NAGATA_1:def.9">Def9</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c9">s</font> being   <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>   <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> <span class="kw"> such that </span><br/><a NAME="E5:51_1_1_1_1_1"/><span class="lab"><font color="Green" title="E33">A6</font></span>: 
<font color="Maroon" title="c9">s</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">and </span><br/><a NAME="E6:51_1_1_1_1_1"/><span class="lab"><font color="Green" title="E34">A7</font></span>: 
 <a href="metric_1.html#K9" title="METRIC_1:func.9">Ball</a> (<font color="Maroon" title="c7">ft</font>,<font color="Maroon" title="c9">s</font>) <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c8">R</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:51_1_1"><span class="lab"><font color="Green" title="E30">A3</font></span></a>, <a class="ref" href="topmetr.html#T15" title="TOPMETR:th.15">TOPMETR:15</a>, <a class="ref" href="topmetr.html#D6" title="TOPMETR:def.6">TOPMETR:def 6</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c10">s9</font> = <font color="Maroon" title="c9">s</font> as   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_0.html#D1" title="XREAL_0:def.1">XREAL_0:def 1</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c11">B</font> =  <a href="metric_1.html#K9" title="METRIC_1:func.9">Ball</a> (<font color="Maroon" title="c7">ft</font>,<font color="Maroon" title="c10">s9</font>) as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <a href="topmetr.html#K3" title="TOPMETR:func.3">R^1</a> <span class="kw">by</span> <span class="lab"><a class="ref" href="metric_1.html#D13" title="METRIC_1:def.13">METRIC_1:def 13</a>, <a class="ref" href="topmetr.html#T17" title="TOPMETR:th.17">TOPMETR:17</a></span>;<br/><a NAME="E9:51_1_1_1_1_1"/>
 <a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> (<font color="Maroon" title="c7">ft</font>,<font color="Maroon" title="c7">ft</font>) <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c10">s9</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:51_1_1_1_1_1"><span class="lab"><font color="Green" title="E33">A6</font></span></a>, <a class="ref" href="metric_1.html#T1" title="METRIC_1:th.1">METRIC_1:1</a></span>;<br/><a NAME="E10:51_1_1_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E35">A8</font></span>: 
<font color="Maroon" title="c7">ft</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">B</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="metric_1.html#T11" title="METRIC_1:th.11">METRIC_1:11</a></span>;<br/><a NAME="E11:51_1_1_1_1_1"/>
( <font color="Maroon" title="c11">B</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  &amp; <font color="Maroon" title="c4">f9</font> <a href="tmap_1.html#R1" title="TMAP_1:pred.1">is_continuous_at</a> <font color="Maroon" title="c6">t</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:51"><span class="lab"><font color="Green" title="E29">A2</font></span></a>, <a class="ref" href="tmap_1.html#T50" title="TMAP_1:th.50">TMAP_1:50</a>, <a class="ref" href="topmetr.html#T14" title="TOPMETR:th.14">TOPMETR:14</a>, <a class="ref" href="topmetr.html#D6" title="TOPMETR:def.6">TOPMETR:def 6</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c12">U</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="E13:51_1_1_1_1_1"/><span class="lab"><font color="Green" title="E36">A9</font></span>: 
( <font color="Maroon" title="c12">U</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  &amp; <font color="Maroon" title="c6">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c12">U</font> )
 <span class="kw">and </span><br/><a NAME="E14:51_1_1_1_1_1"/><span class="lab"><font color="Green" title="E37">A10</font></span>: 
<font color="Maroon" title="c4">f9</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c12">U</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c11">B</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:51_1_1_1_1_1"><span class="lab"><font color="Green" title="E35">A8</font></span></a>, <a class="ref" href="tmap_1.html#T43" title="TMAP_1:th.43">TMAP_1:43</a></span>;<br/><a NAME="E15:51_1_1_1_1_1"/>
<span class="p1">(<span class="default"><a href="nagata_1.html#K5" title="NAGATA_1:func.5">min</a> (<font color="Maroon" title="c2">r</font>,<font color="Maroon" title="c3">f</font>)</span>)</span> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c12">U</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c8">R</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="51_1_1_1_1_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c13">mfx</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c13">mfx</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="nagata_1.html#K5" title="NAGATA_1:func.5">min</a> (<font color="Maroon" title="c2">r</font>,<font color="Maroon" title="c3">f</font>)</span>)</span> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c12">U</font> or <font color="Maroon" title="c13">mfx</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">R</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:51_1_1_1_1_1_1"/>
<font color="Maroon" title="c13">mfx</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="nagata_1.html#K5" title="NAGATA_1:func.5">min</a> (<font color="Maroon" title="c2">r</font>,<font color="Maroon" title="c3">f</font>)</span>)</span> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c12">U</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c13">mfx</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">R</font></span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c14">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E3:51_1_1_1_1_1_1"/><span class="lab"><font color="Green" title="E38">A11</font></span>: 
<font color="Maroon" title="c14">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="nagata_1.html#K5" title="NAGATA_1:func.5">min</a> (<font color="Maroon" title="c2">r</font>,<font color="Maroon" title="c3">f</font>)</span>)</span>
 <span class="kw">and </span><br/><a NAME="E4:51_1_1_1_1_1_1"/><span class="lab"><font color="Green" title="E39">A12</font></span>: 
<font color="Maroon" title="c14">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c12">U</font>
 <span class="kw">and </span><br/><a NAME="E5:51_1_1_1_1_1_1"/><span class="lab"><font color="Green" title="E40">A13</font></span>: 
<font color="Maroon" title="c13">mfx</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="nagata_1.html#K5" title="NAGATA_1:func.5">min</a> (<font color="Maroon" title="c2">r</font>,<font color="Maroon" title="c3">f</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">x</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#D6" title="FUNCT_1:def.6">FUNCT_1:def 6</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c15">x</font> = <font color="Maroon" title="c14">x</font> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">T</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:51_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E38">A11</font></span></a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c16">fx</font> = <font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c15">x</font>, <font color="Maroon" title="c17">r9</font> = <font color="Maroon" title="c2">r</font> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <a href="metric_1.html#K8" title="METRIC_1:func.8">RealSpace</a> <span class="kw">by</span> <span class="lab"><a class="ref" href="metric_1.html#D13" title="METRIC_1:def.13">METRIC_1:def 13</a></span>;<br/>
<a NAME="E8:51_1_1_1_1_1_1"/>
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="nagata_1.html#K5" title="NAGATA_1:func.5">min</a> (<font color="Maroon" title="c2">r</font>,<font color="Maroon" title="c3">f</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></span>;<br/>
<a NAME="E9:51_1_1_1_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c15">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c3">f</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:51_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E38">A11</font></span></a>, <a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></span>;<br/>
<a NAME="E10:51_1_1_1_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E41">A14</font></span>: 
<font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c15">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">f</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c12">U</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:51_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E39">A12</font></span></a>, <a class="ref" href="funct_1.html#D6" title="FUNCT_1:def.6">FUNCT_1:def 6</a></span>;<br/>
<a NAME="E11:51_1_1_1_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E42">A15</font></span>: 
<font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c15">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">B</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:51_1_1_1_1_1"><span class="lab"><font color="Green" title="E37">A10</font></span></a></span>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="51_1_1_1_1_1_1_1"><span class="kw">now </span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="51_1_1_1_1_1_1_1_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:51_1_1_1_1_1_1_1_1"/>
( <font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c15">x</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">r</font> or <font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c15">x</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c2">r</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="51_1_1_1_1_1_1_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:51_1_1_1_1_1_1_1_1_1"/>
<font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c15">x</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">r</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c13">mfx</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">R</font></span><br/><div class="add"><a NAME="E2:51_1_1_1_1_1_1_1_1_1"/><span class="kw">then </span>
 <a href="xxreal_0.html#K3" title="XXREAL_0:func.3">min</a> (<font color="Maroon" title="c2">r</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c15">x</font></span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c15">x</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_0.html#D9" title="XXREAL_0:def.9">XXREAL_0:def 9</a></span>;<br/><a NAME="E3:51_1_1_1_1_1_1_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c13">mfx</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c15">x</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:51_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E40">A13</font></span></a>, <a class="ref" href="nagata_1.html#D9" target="_self" title="NAGATA_1:def.9">Def9</a></span>;<br/><span class="kw">hence </span><a NAME="E4:51_1_1_1_1_1_1_1_1_1"/>
<font color="Maroon" title="c13">mfx</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">R</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:51_1_1_1_1_1"><span class="lab"><font color="Green" title="E34">A7</font></span></a>, <a class="txt" href="#E11:51_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E42">A15</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="51_1_1_1_1_1_1_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:51_1_1_1_1_1_1_1_1_2"/><span class="lab"><font color="Green" title="E43">A16</font></span>: 
<font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c15">x</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c2">r</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c13">mfx</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">R</font></span><br/><div class="add"><a NAME="E2:51_1_1_1_1_1_1_1_1_2"/>
 <a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> (<font color="Maroon" title="c16">fx</font>,<font color="Maroon" title="c7">ft</font>) <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c9">s</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:51_1_1_1_1_1"><span class="lab"><font color="Green" title="E37">A10</font></span></a>, <a class="txt" href="#E10:51_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E41">A14</font></span></a>, <a class="ref" href="metric_1.html#T11" title="METRIC_1:th.11">METRIC_1:11</a></span>;<br/><a NAME="E3:51_1_1_1_1_1_1_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E44">A17</font></span>: 
 <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c15">x</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">t</font></span>)</span></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c9">s</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="topmetr.html#T11" title="TOPMETR:th.11">TOPMETR:11</a></span>;<br/><a NAME="E4:51_1_1_1_1_1_1_1_1_2"/><span class="lab"><font color="Green" title="E45">A18</font></span>: 
<font color="Maroon" title="c2">r</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">t</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c15">x</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">t</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:51_1_1_1_1_1_1_1_1_2"><span class="lab"><font color="Green" title="E43">A16</font></span></a>, <a class="ref" href="xreal_1.html#T9" title="XREAL_1:th.9">XREAL_1:9</a></span>;<br/><a NAME="E5:51_1_1_1_1_1_1_1_1_2"/>
<font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c15">x</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">t</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:51_1_1_1_1_1"><span class="lab"><font color="Green" title="E32">A5</font></span></a>, <a class="txt" href="#E1:51_1_1_1_1_1_1_1_1_2"><span class="lab"><font color="Green" title="E43">A16</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><a NAME="E6:51_1_1_1_1_1_1_1_1_2"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c15">x</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">t</font></span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T48" title="XREAL_1:th.48">XREAL_1:48</a></span>;<br/><a NAME="E7:51_1_1_1_1_1_1_1_1_2"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c15">x</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">t</font></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c9">s</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:51_1_1_1_1_1_1_1_1_2"><span class="lab"><font color="Green" title="E44">A17</font></span></a>, <a class="ref" href="absvalue.html#D1" title="ABSVALUE:def.1">ABSVALUE:def 1</a></span>;<br/><a NAME="E8:51_1_1_1_1_1_1_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E46">A19</font></span>: 
<font color="Maroon" title="c2">r</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">t</font></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c9">s</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:51_1_1_1_1_1_1_1_1_2"><span class="lab"><font color="Green" title="E45">A18</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><a NAME="E9:51_1_1_1_1_1_1_1_1_2"/>
<font color="Maroon" title="c2">r</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">t</font></span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:51_1_1_1_1_1"><span class="lab"><font color="Green" title="E32">A5</font></span></a>, <a class="ref" href="xreal_1.html#T48" title="XREAL_1:th.48">XREAL_1:48</a></span>;<br/><a NAME="E10:51_1_1_1_1_1_1_1_1_2"/><span class="kw">then </span>
 <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">r</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">t</font></span>)</span></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c9">s</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:51_1_1_1_1_1_1_1_1_2"><span class="lab"><font color="Green" title="E46">A19</font></span></a>, <a class="ref" href="absvalue.html#D1" title="ABSVALUE:def.1">ABSVALUE:def 1</a></span>;<br/><a NAME="E11:51_1_1_1_1_1_1_1_1_2"/><span class="kw">then </span>
 <a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> (<font color="Maroon" title="c7">ft</font>,<font color="Maroon" title="c17">r9</font>) <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c9">s</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="topmetr.html#T11" title="TOPMETR:th.11">TOPMETR:11</a></span>;<br/><a NAME="E12:51_1_1_1_1_1_1_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E47">A20</font></span>: 
<font color="Maroon" title="c2">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">B</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="metric_1.html#T11" title="METRIC_1:th.11">METRIC_1:11</a></span>;<br/><a NAME="E13:51_1_1_1_1_1_1_1_1_2"/>
 <a href="xxreal_0.html#K3" title="XXREAL_0:func.3">min</a> (<font color="Maroon" title="c2">r</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c15">x</font></span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">r</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:51_1_1_1_1_1_1_1_1_2"><span class="lab"><font color="Green" title="E43">A16</font></span></a>, <a class="ref" href="xxreal_0.html#D9" title="XXREAL_0:def.9">XXREAL_0:def 9</a></span>;<br/><a NAME="E14:51_1_1_1_1_1_1_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c13">mfx</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">r</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:51_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E40">A13</font></span></a>, <a class="ref" href="nagata_1.html#D9" target="_self" title="NAGATA_1:def.9">Def9</a></span>;<br/><span class="kw">hence </span><a NAME="E15:51_1_1_1_1_1_1_1_1_2"/>
<font color="Maroon" title="c13">mfx</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">R</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:51_1_1_1_1_1"><span class="lab"><font color="Green" title="E34">A7</font></span></a>, <a class="txt" href="#E12:51_1_1_1_1_1_1_1_1_2"><span class="lab"><font color="Green" title="E47">A20</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E13:51_1_1_1_1_1_1"/>
<font color="Maroon" title="c13">mfx</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">R</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E16:51_1_1_1_1_1"/>
 ex <font color="Olive" title="b1">U</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">T</font> st <br/>( <font color="Olive" title="b1">U</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  &amp; <font color="Maroon" title="c6">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">U</font> &amp; <span class="p1">(<span class="default"><a href="nagata_1.html#K5" title="NAGATA_1:func.5">min</a> (<font color="Maroon" title="c2">r</font>,<font color="Maroon" title="c3">f</font>)</span>)</span> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Olive" title="b1">U</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c8">R</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:51_1_1_1_1_1"><span class="lab"><font color="Green" title="E36">A9</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="51_1_1_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:51_1_1_1_1_2"/><span class="lab"><font color="Green" title="E32">A21</font></span>: 
<font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">t</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c2">r</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">U</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">T</font> st <br/>( <font color="Olive" title="b1">U</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  &amp; <font color="Maroon" title="c6">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">U</font> &amp; <span class="p1">(<span class="default"><a href="nagata_1.html#K5" title="NAGATA_1:func.5">min</a> (<font color="Maroon" title="c2">r</font>,<font color="Maroon" title="c3">f</font>)</span>)</span> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Olive" title="b1">U</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c8">R</font> )</span><br/><div class="add"><span class="kw">set </span><font color="Maroon" title="c9">s</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">t</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">r</font>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c10">B</font> =  <a href="metric_1.html#K9" title="METRIC_1:func.9">Ball</a> (<font color="Maroon" title="c7">ft</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">t</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">r</font></span>)</span>) as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <a href="topmetr.html#K3" title="TOPMETR:func.3">R^1</a> <span class="kw">by</span> <span class="lab"><a class="ref" href="metric_1.html#D13" title="METRIC_1:def.13">METRIC_1:def 13</a>, <a class="ref" href="topmetr.html#T17" title="TOPMETR:th.17">TOPMETR:17</a></span>;<br/><a NAME="E3:51_1_1_1_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">t</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">r</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:51_1_1_1_1_2"><span class="lab"><font color="Green" title="E32">A21</font></span></a>, <a class="ref" href="xreal_1.html#T50" title="XREAL_1:th.50">XREAL_1:50</a></span>;<br/><a NAME="E4:51_1_1_1_1_2"/><span class="kw">then </span>
 <a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> (<font color="Maroon" title="c7">ft</font>,<font color="Maroon" title="c7">ft</font>) <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">t</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">r</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="metric_1.html#T1" title="METRIC_1:th.1">METRIC_1:1</a></span>;<br/><a NAME="E5:51_1_1_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E33">A22</font></span>: 
<font color="Maroon" title="c7">ft</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">B</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="metric_1.html#T11" title="METRIC_1:th.11">METRIC_1:11</a></span>;<br/><a NAME="E6:51_1_1_1_1_2"/>
( <font color="Maroon" title="c10">B</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  &amp; <font color="Maroon" title="c4">f9</font> <a href="tmap_1.html#R1" title="TMAP_1:pred.1">is_continuous_at</a> <font color="Maroon" title="c6">t</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:51"><span class="lab"><font color="Green" title="E29">A2</font></span></a>, <a class="ref" href="tmap_1.html#T50" title="TMAP_1:th.50">TMAP_1:50</a>, <a class="ref" href="topmetr.html#T14" title="TOPMETR:th.14">TOPMETR:14</a>, <a class="ref" href="topmetr.html#D6" title="TOPMETR:def.6">TOPMETR:def 6</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c11">U</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="E8:51_1_1_1_1_2"/><span class="lab"><font color="Green" title="E34">A23</font></span>: 
( <font color="Maroon" title="c11">U</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  &amp; <font color="Maroon" title="c6">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">U</font> )
 <span class="kw">and </span><br/><a NAME="E9:51_1_1_1_1_2"/><span class="lab"><font color="Green" title="E35">A24</font></span>: 
<font color="Maroon" title="c4">f9</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c11">U</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c10">B</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:51_1_1_1_1_2"><span class="lab"><font color="Green" title="E33">A22</font></span></a>, <a class="ref" href="tmap_1.html#T43" title="TMAP_1:th.43">TMAP_1:43</a></span>;<br/><a NAME="E10:51_1_1_1_1_2"/>
<span class="p1">(<span class="default"><a href="nagata_1.html#K5" title="NAGATA_1:func.5">min</a> (<font color="Maroon" title="c2">r</font>,<font color="Maroon" title="c3">f</font>)</span>)</span> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c11">U</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c8">R</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="51_1_1_1_1_2_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c12">mfx</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c12">mfx</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="nagata_1.html#K5" title="NAGATA_1:func.5">min</a> (<font color="Maroon" title="c2">r</font>,<font color="Maroon" title="c3">f</font>)</span>)</span> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c11">U</font> or <font color="Maroon" title="c12">mfx</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">R</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:51_1_1_1_1_2_1"/>
<font color="Maroon" title="c12">mfx</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="nagata_1.html#K5" title="NAGATA_1:func.5">min</a> (<font color="Maroon" title="c2">r</font>,<font color="Maroon" title="c3">f</font>)</span>)</span> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c11">U</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c12">mfx</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">R</font></span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c13">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E3:51_1_1_1_1_2_1"/><span class="lab"><font color="Green" title="E36">A25</font></span>: 
<font color="Maroon" title="c13">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="nagata_1.html#K5" title="NAGATA_1:func.5">min</a> (<font color="Maroon" title="c2">r</font>,<font color="Maroon" title="c3">f</font>)</span>)</span>
 <span class="kw">and </span><br/><a NAME="E4:51_1_1_1_1_2_1"/><span class="lab"><font color="Green" title="E37">A26</font></span>: 
<font color="Maroon" title="c13">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">U</font>
 <span class="kw">and </span><br/><a NAME="E5:51_1_1_1_1_2_1"/><span class="lab"><font color="Green" title="E38">A27</font></span>: 
<font color="Maroon" title="c12">mfx</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="nagata_1.html#K5" title="NAGATA_1:func.5">min</a> (<font color="Maroon" title="c2">r</font>,<font color="Maroon" title="c3">f</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">x</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#D6" title="FUNCT_1:def.6">FUNCT_1:def 6</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c14">x</font> = <font color="Maroon" title="c13">x</font> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">T</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:51_1_1_1_1_2_1"><span class="lab"><font color="Green" title="E36">A25</font></span></a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c15">fx</font> = <font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c14">x</font> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <a href="metric_1.html#K8" title="METRIC_1:func.8">RealSpace</a> <span class="kw">by</span> <span class="lab"><a class="ref" href="metric_1.html#D13" title="METRIC_1:def.13">METRIC_1:def 13</a></span>;<br/>
<a NAME="E8:51_1_1_1_1_2_1"/>
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="nagata_1.html#K5" title="NAGATA_1:func.5">min</a> (<font color="Maroon" title="c2">r</font>,<font color="Maroon" title="c3">f</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></span>;<br/>
<a NAME="E9:51_1_1_1_1_2_1"/><span class="kw">then </span>
<font color="Maroon" title="c14">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c3">f</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:51_1_1_1_1_2_1"><span class="lab"><font color="Green" title="E36">A25</font></span></a>, <a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></span>;<br/>
<a NAME="E10:51_1_1_1_1_2_1"/><span class="kw">then </span>
<font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c14">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">f</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c11">U</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:51_1_1_1_1_2_1"><span class="lab"><font color="Green" title="E37">A26</font></span></a>, <a class="ref" href="funct_1.html#D6" title="FUNCT_1:def.6">FUNCT_1:def 6</a></span>;<br/>
<a NAME="E11:51_1_1_1_1_2_1"/><span class="kw">then </span>
 <a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> (<font color="Maroon" title="c7">ft</font>,<font color="Maroon" title="c15">fx</font>) <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">t</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">r</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:51_1_1_1_1_2"><span class="lab"><font color="Green" title="E35">A24</font></span></a>, <a class="ref" href="metric_1.html#T11" title="METRIC_1:th.11">METRIC_1:11</a></span>;<br/>
<a NAME="E12:51_1_1_1_1_2_1"/><span class="kw">then </span>
 <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">t</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c14">x</font></span>)</span></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">t</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">r</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="topmetr.html#T11" title="TOPMETR:th.11">TOPMETR:11</a></span>;<br/>
<a NAME="E13:51_1_1_1_1_2_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">t</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c14">x</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">t</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> <font color="Maroon" title="c2">r</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="absvalue.html#T5" title="ABSVALUE:th.5">ABSVALUE:5</a></span>;<br/>
<a NAME="E14:51_1_1_1_1_2_1"/><span class="kw">then </span>
 <a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c14">x</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> <font color="Maroon" title="c2">r</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T6" title="XREAL_1:th.6">XREAL_1:6</a></span>;<br/>
<a NAME="E15:51_1_1_1_1_2_1"/><span class="kw">then </span>
<font color="Maroon" title="c2">r</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c14">x</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T24" title="XREAL_1:th.24">XREAL_1:24</a></span>;<br/>
<a NAME="E16:51_1_1_1_1_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E39">A28</font></span>: 
 <a href="xxreal_0.html#K3" title="XXREAL_0:func.3">min</a> (<font color="Maroon" title="c2">r</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c14">x</font></span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">r</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_0.html#D9" title="XXREAL_0:def.9">XXREAL_0:def 9</a></span>;<br/>
<a NAME="E17:51_1_1_1_1_2_1"/>
 <a href="xxreal_0.html#K3" title="XXREAL_0:func.3">min</a> (<font color="Maroon" title="c2">r</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">t</font></span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">r</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:51_1_1_1_1_2"><span class="lab"><font color="Green" title="E32">A21</font></span></a>, <a class="ref" href="xxreal_0.html#D9" title="XXREAL_0:def.9">XXREAL_0:def 9</a></span>;<br/>
<a NAME="E18:51_1_1_1_1_2_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="nagata_1.html#K5" title="NAGATA_1:func.5">min</a> (<font color="Maroon" title="c2">r</font>,<font color="Maroon" title="c3">f</font>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">r</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nagata_1.html#D9" target="_self" title="NAGATA_1:def.9">Def9</a></span>;<br/>
<span class="kw">hence </span><a NAME="E19:51_1_1_1_1_2_1"/>
<font color="Maroon" title="c12">mfx</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">R</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:51_1_1"><span class="lab"><font color="Green" title="E31">A4</font></span></a>, <a class="txt" href="#E5:51_1_1_1_1_2_1"><span class="lab"><font color="Green" title="E38">A27</font></span></a>, <a class="txt" href="#E16:51_1_1_1_1_2_1"><span class="lab"><font color="Green" title="E39">A28</font></span></a>, <a class="ref" href="nagata_1.html#D9" target="_self" title="NAGATA_1:def.9">Def9</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E11:51_1_1_1_1_2"/>
 ex <font color="Olive" title="b1">U</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">T</font> st <br/>( <font color="Olive" title="b1">U</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  &amp; <font color="Maroon" title="c6">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">U</font> &amp; <span class="p1">(<span class="default"><a href="nagata_1.html#K5" title="NAGATA_1:func.5">min</a> (<font color="Maroon" title="c2">r</font>,<font color="Maroon" title="c3">f</font>)</span>)</span> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Olive" title="b1">U</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c8">R</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:51_1_1_1_1_2"><span class="lab"><font color="Green" title="E34">A23</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E5:51_1_1"/>
 ex <font color="Olive" title="b1">U</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">T</font> st <br/>( <font color="Olive" title="b1">U</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  &amp; <font color="Maroon" title="c6">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">U</font> &amp; <font color="Maroon" title="c5">mf9</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Olive" title="b1">U</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c8">R</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E2:51_1"/>
<font color="Maroon" title="c5">mf9</font> <a href="tmap_1.html#R1" title="TMAP_1:pred.1">is_continuous_at</a> <font color="Maroon" title="c6">t</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tmap_1.html#T43" title="TMAP_1:th.43">TMAP_1:43</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<a NAME="E5:51"/><span class="kw">then </span>
<font color="Maroon" title="c5">mf9</font> is  <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="tmap_1.html#T50" title="TMAP_1:th.50">TMAP_1:50</a></span>;<br/>
<span class="kw">hence </span><a NAME="E6:51"/>
 <a href="nagata_1.html#K5" title="NAGATA_1:func.5">min</a> (<font color="Maroon" title="c2">r</font>,<font color="Maroon" title="c3">f</font>) is  <a href="pscomp_1.html#V1" title="PSCOMP_1:attr.1">continuous</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="topreal6.html#T74" title="TOPREAL6:th.74">TOPREAL6:74</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
