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<b>let </b><font color="Maroon">f</font> be   <a href="struct_0.html#NM6">Function</a> of <span class="p1">(<span class="default"><a href="pcomps_1.html#K4">TopSpaceMetr</a> <font color="Maroon">M</font></span>)</span>,<span class="p1">(<span class="default"><a href="pcomps_1.html#K4">TopSpaceMetr</a> <font color="Maroon">M</font></span>)</span>;<br/>

<b>assume </b><a NAME="E1:6_1_1"/><i><font color="Green">E49</font></i>: 
<font color="Maroon">f</font> is <a href="jordan24.html#V1" target="_self">isometric</a>
 ;<br/>

<b>then consider </b><font color="Maroon">g</font> being  <a href="vectmetr.html#V3">isometric</a> <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">M</font>,<font color="Maroon">M</font><b> such that </b><br/><a NAME="E3:6_1_1"/><i><font color="Green">E50</font></i>: 
<font color="Maroon">g</font> <a href="funct_2.html#R1">=</a> <font color="Maroon">f</font>
 <b>by </b><i/>;<br/>
<b>thus </b><a NAME="E4:6_1_1"/>
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#K2">[#]</a> <span class="p1">(<span class="default"><a href="pcomps_1.html#K4">TopSpaceMetr</a> <font color="Maroon">M</font></span>)</span>
 <b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="tops_2.html#D5">TOPS_2:def 5</a><br/>

<a NAME="E5:6_1_1"/>
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">g</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">M</font>
 
<b>by </b><i><a class="ref" href="vectmetr.html#D3">VECTMETR:def 3</a></i>;<br/>
<b>hence </b><a NAME="E6:6_1_1"/>
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#K2">[#]</a> <span class="p1">(<span class="default"><a href="pcomps_1.html#K4">TopSpaceMetr</a> <font color="Maroon">M</font></span>)</span>
 <b>by </b><i/>;<br/>

<b>thus </b><a NAME="E7:6_1_1"/>
<font color="Maroon">f</font> is <a href="funct_1.html#V2">one-to-one</a>
 <b>by </b><i/>;<br/>

<b>reconsider </b><font color="Maroon">f1</font> = <font color="Maroon">f</font> as  <a href="jordan24.html#V1" target="_self">isometric</a> <a href="struct_0.html#NM6">Function</a> of <span class="p1">(<span class="default"><a href="pcomps_1.html#K4">TopSpaceMetr</a> <font color="Maroon">M</font></span>)</span>,<span class="p1">(<span class="default"><a href="pcomps_1.html#K4">TopSpaceMetr</a> <font color="Maroon">M</font></span>)</span> <b>by </b><i/>;<br/>
<a NAME="E9:6_1_1"/>
<font color="Maroon">f1</font> is <a href="pre_topc.html#V5">continuous</a>
 
;<br/>
<b>hence </b><a NAME="E10:6_1_1"/>
<font color="Maroon">f</font> is <a href="pre_topc.html#V5">continuous</a>
 ;<br/>

<a NAME="E11:6_1_1"/><i><font color="Green">E66</font></i>: 
<font color="Maroon">g</font> <a href="tops_2.html#NK3" target="_self">"</a>  is  <a href="vectmetr.html#V3">isometric</a> <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">M</font>,<font color="Maroon">M</font>
 
<b>by </b><i><a class="ref" href="vectmetr.html#T3">VECTMETR:3</a></i>;<br/>
<b>reconsider </b><font color="Maroon">g1</font> = <font color="Maroon">g</font> <a href="tops_2.html#NK3" target="_self">"</a>  as   <a href="struct_0.html#NM6">Function</a> of <span class="p1">(<span class="default"><a href="pcomps_1.html#K4">TopSpaceMetr</a> <font color="Maroon">M</font></span>)</span>,<span class="p1">(<span class="default"><a href="pcomps_1.html#K4">TopSpaceMetr</a> <font color="Maroon">M</font></span>)</span> ;<br/>
<a NAME="E13:6_1_1"/>
<font color="Maroon">f</font> <a href="tops_2.html#NK3" target="_self">"</a>  <a href="funct_2.html#R1">=</a> <font color="Maroon">g</font> <a href="tops_2.html#NK3" target="_self">"</a> 
 
<b>by </b><i/>;<br/>
<a NAME="E14:6_1_1"/><b>then </b>
<font color="Maroon">f</font> <a href="tops_2.html#NK3" target="_self">"</a>  is  <a href="jordan24.html#V1" target="_self">isometric</a> <a href="struct_0.html#NM6">Function</a> of <span class="p1">(<span class="default"><a href="pcomps_1.html#K4">TopSpaceMetr</a> <font color="Maroon">M</font></span>)</span>,<span class="p1">(<span class="default"><a href="pcomps_1.html#K4">TopSpaceMetr</a> <font color="Maroon">M</font></span>)</span>
 
<b>by </b><i>, </i>;<br/>
<b>hence </b><a NAME="E15:6_1_1"/>
<font color="Maroon">f</font> <a href="tops_2.html#NK3" target="_self">"</a>  is <a href="pre_topc.html#V5">continuous</a>
 ;<br/>


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