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

<b>let </b><font color="Maroon" title="c6">f</font> be   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</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="c5">M</font></span>)</span>,<span class="p1">(<span class="default"><a href="pcomps_1.html#K4" title="PCOMPS_1:func.4">TopSpaceMetr</a> <font color="Maroon" title="c5">M</font></span>)</span>;<br/>

<b>assume </b><a NAME="E1:6_1_1"/><i><font color="Green" title="E4">A1</font></i>: 
<font color="Maroon" title="c6">f</font> is <a href="jordan24.html#V1" title="JORDAN24:attr.1">isometric</a>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c7">g</font> being  <a href="vectmetr.html#V3" title="VECTMETR:attr.3">isometric</a> <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c5">M</font>,<font color="Maroon" title="c5">M</font><b> such that </b><br/><a NAME="E3:6_1_1"/><i><font color="Green" title="E5">A2</font></i>: 
<font color="Maroon" title="c7">g</font> <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a> <font color="Maroon" title="c6">f</font>
 <b>by </b><i><a class="ref" href="jordan24.html#D2" target="_self" title="JORDAN24:def.2">Def2</a></i>;<br/>
<b>thus </b><a NAME="E4:6_1_1"/>
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c6">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p1">(<span class="default"><a href="pcomps_1.html#K4" title="PCOMPS_1:func.4">TopSpaceMetr</a> <font color="Maroon" title="c5">M</font></span>)</span>
 <b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="tops_2.html#D5" title="TOPS_2:def.5">TOPS_2:def 5</a><br/>

<a NAME="E5:6_1_1"/>
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c7">g</font> <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="c5">M</font>
 
<b>by </b><i><a class="ref" href="vectmetr.html#D3" title="VECTMETR:def.3">VECTMETR:def 3</a></i>;<br/>
<b>hence </b><a NAME="E6:6_1_1"/>
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c6">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p1">(<span class="default"><a href="pcomps_1.html#K4" title="PCOMPS_1:func.4">TopSpaceMetr</a> <font color="Maroon" title="c5">M</font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan24.html#E3:6_1_1"><i><font color="Green" title="E5">A2</font></i></a></i>;<br/>

<b>thus </b><a NAME="E7:6_1_1"/>
<font color="Maroon" title="c6">f</font> is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>
 <b>by </b><i><a class="txt" href="jordan24.html#E3:6_1_1"><i><font color="Green" title="E5">A2</font></i></a></i>;<br/>

<b>reconsider </b><font color="Maroon" title="c8">f1</font> = <font color="Maroon" title="c6">f</font> as  <a href="jordan24.html#V1" title="JORDAN24:attr.1">isometric</a> <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</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="c5">M</font></span>)</span>,<span class="p1">(<span class="default"><a href="pcomps_1.html#K4" title="PCOMPS_1:func.4">TopSpaceMetr</a> <font color="Maroon" title="c5">M</font></span>)</span> <b>by </b><i><a class="txt" href="jordan24.html#E1:6_1_1"><i><font color="Green" title="E4">A1</font></i></a></i>;<br/>
<a NAME="E9:6_1_1"/>
<font color="Maroon" title="c8">f1</font> is <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a>
 
;<br/>
<b>hence </b><a NAME="E10:6_1_1"/>
<font color="Maroon" title="c6">f</font> is <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a>
 ;<br/>

<a NAME="E11:6_1_1"/><i><font color="Green" title="E6">A3</font></i>: 
<font color="Maroon" title="c7">g</font> <a href="tops_2.html#NK3" title="TOPS_2:NK.3">"</a>  is  <a href="vectmetr.html#V3" title="VECTMETR:attr.3">isometric</a> <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c5">M</font>,<font color="Maroon" title="c5">M</font>
 
<b>by </b><i><a class="ref" href="vectmetr.html#T3" title="VECTMETR:th.3">VECTMETR:3</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c9">g1</font> = <font color="Maroon" title="c7">g</font> <a href="tops_2.html#NK3" title="TOPS_2:NK.3">"</a>  as   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</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="c5">M</font></span>)</span>,<span class="p1">(<span class="default"><a href="pcomps_1.html#K4" title="PCOMPS_1:func.4">TopSpaceMetr</a> <font color="Maroon" title="c5">M</font></span>)</span> ;<br/>
<a NAME="E13:6_1_1"/>
<font color="Maroon" title="c6">f</font> <a href="tops_2.html#NK3" title="TOPS_2:NK.3">"</a>  <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a> <font color="Maroon" title="c7">g</font> <a href="tops_2.html#NK3" title="TOPS_2:NK.3">"</a> 
 
<b>by </b><i><a class="txt" href="jordan24.html#E3:6_1_1"><i><font color="Green" title="E5">A2</font></i></a></i>;<br/>
<a NAME="E14:6_1_1"/><b>then </b>
<font color="Maroon" title="c6">f</font> <a href="tops_2.html#NK3" title="TOPS_2:NK.3">"</a>  is  <a href="jordan24.html#V1" title="JORDAN24:attr.1">isometric</a> <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</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="c5">M</font></span>)</span>,<span class="p1">(<span class="default"><a href="pcomps_1.html#K4" title="PCOMPS_1:func.4">TopSpaceMetr</a> <font color="Maroon" title="c5">M</font></span>)</span>
 
<b>by </b><i><a class="ref" href="jordan24.html#D2" target="_self" title="JORDAN24:def.2">Def2</a>, <a class="txt" href="jordan24.html#E11:6_1_1"><i><font color="Green" title="E6">A3</font></i></a></i>;<br/>
<b>hence </b><a NAME="E15:6_1_1"/>
<font color="Maroon" title="c6">f</font> <a href="tops_2.html#NK3" title="TOPS_2:NK.3">"</a>  is <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a>
 ;<br/>


</div>
