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<b>let </b><font color="Maroon" title="c4">P</font>, <font color="Maroon" title="c5">Q</font> be  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="compts_1.html#V2" title="COMPTS_1:attr.2">compact</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="pcomps_1.html#K4" title="PCOMPS_1:func.4">TopSpaceMetr</a> <font color="Maroon" title="c1">M</font></span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="weierstr.html#K10" title="WEIERSTR:func.10">max_dist_max</a> <font color="Maroon" title="c4">P</font>,<font color="Maroon" title="c5">Q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="weierstr.html#K10" title="WEIERSTR:func.10">max_dist_max</a> <font color="Maroon" title="c5">Q</font>,<font color="Maroon" title="c4">P</font></span><br/>

<b>consider </b><font color="Maroon" title="c6">y1</font>, <font color="Maroon" title="c7">y2</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">M</font><b> such that </b><br/><a NAME="E2:39_2_1"/><i><font color="Green" title="E30">A8</font></i>: 
<font color="Maroon" title="c6">y1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">Q</font>
 <b>and </b><br/><a NAME="E3:39_2_1"/><i><font color="Green" title="E31">A9</font></i>: 
<font color="Maroon" title="c7">y2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">P</font>
 <b>and </b><br/><a NAME="E4:39_2_1"/><i><font color="Green" title="E32">A10</font></i>: 
 <a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c6">y1</font>,<font color="Maroon" title="c7">y2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="weierstr.html#K10" title="WEIERSTR:func.10">max_dist_max</a> <font color="Maroon" title="c5">Q</font>,<font color="Maroon" title="c4">P</font>
 <b>by </b><i><a class="ref" href="weierstr.html#T39" title="WEIERSTR:th.39">WEIERSTR:39</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c8">x1</font>, <font color="Maroon" title="c9">x2</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">M</font><b> such that </b><br/><a NAME="E6:39_2_1"/><i><font color="Green" title="E33">A11</font></i>: 
<font color="Maroon" title="c8">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">P</font>
 <b>and </b><br/><a NAME="E7:39_2_1"/><i><font color="Green" title="E34">A12</font></i>: 
<font color="Maroon" title="c9">x2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">Q</font>
 <b>and </b><br/><a NAME="E8:39_2_1"/><i><font color="Green" title="E35">A13</font></i>: 
 <a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c8">x1</font>,<font color="Maroon" title="c9">x2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="weierstr.html#K10" title="WEIERSTR:func.10">max_dist_max</a> <font color="Maroon" title="c4">P</font>,<font color="Maroon" title="c5">Q</font>
 <b>by </b><i><a class="ref" href="weierstr.html#T39" title="WEIERSTR:th.39">WEIERSTR:39</a></i>;<br/>
<a NAME="E9:39_2_1"/><i><font color="Green" title="E36">A14</font></i>: 
 <a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c8">x1</font>,<font color="Maroon" title="c9">x2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c6">y1</font>,<font color="Maroon" title="c7">y2</font>
 
<b>by </b><i><a class="txt" href="#E4:39_2_1"><i><font color="Green" title="E32">A10</font></i></a>, <a class="txt" href="#E6:39_2_1"><i><font color="Green" title="E33">A11</font></i></a>, <a class="txt" href="#E7:39_2_1"><i><font color="Green" title="E34">A12</font></i></a>, <a class="ref" href="weierstr.html#T40" title="WEIERSTR:th.40">WEIERSTR:40</a></i>;<br/>
<a NAME="E10:39_2_1"/>
 <a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c6">y1</font>,<font color="Maroon" title="c7">y2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c8">x1</font>,<font color="Maroon" title="c9">x2</font>
 
<b>by </b><i><a class="txt" href="#E2:39_2_1"><i><font color="Green" title="E30">A8</font></i></a>, <a class="txt" href="#E3:39_2_1"><i><font color="Green" title="E31">A9</font></i></a>, <a class="txt" href="#E8:39_2_1"><i><font color="Green" title="E35">A13</font></i></a>, <a class="ref" href="weierstr.html#T40" title="WEIERSTR:th.40">WEIERSTR:40</a></i>;<br/>
<b>hence </b><a NAME="E11:39_2_1"/>
 <a href="weierstr.html#K10" title="WEIERSTR:func.10">max_dist_max</a> <font color="Maroon" title="c4">P</font>,<font color="Maroon" title="c5">Q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="weierstr.html#K10" title="WEIERSTR:func.10">max_dist_max</a> <font color="Maroon" title="c5">Q</font>,<font color="Maroon" title="c4">P</font>
 <b>by </b><i><a class="txt" href="#E4:39_2_1"><i><font color="Green" title="E32">A10</font></i></a>, <a class="txt" href="#E8:39_2_1"><i><font color="Green" title="E35">A13</font></i></a>, <a class="txt" href="#E9:39_2_1"><i><font color="Green" title="E36">A14</font></i></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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