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<b>let </b><font color="Maroon">c<sub>6</sub></font> be   <a href="struct_0.html#NM7">Function</a> of <span class="p1">(<span class="default"><a href="pcomps_1.html#K4">TopSpaceMetr</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="pcomps_1.html#K4">TopSpaceMetr</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>;<br/>

<b>assume </b><a NAME="E1:6_1_1"/><i><font color="Green">E4</font></i>: 
<font color="Maroon">c<sub>6</sub></font> is <a href="jordan24.html#V1" target="_self">isometric</a>
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>7</sub></font> being  <a href="vectmetr.html#V3">isometric</a> <a href="struct_0.html#NM7">Function</a> of <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>5</sub></font><b> such that </b><br/><a NAME="E3:6_1_1"/><i><font color="Green">E5</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="funct_2.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="ref" href="jordan24.html#D2" target="_self">Def2</a></i>;<br/>
<b>thus </b> <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>6</sub></font> = 
the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="pcomps_1.html#K4">TopSpaceMetr</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>
<br/>.= 
 <a href="pre_topc.html#K2">[#]</a> <span class="p1">(<span class="default"><a href="pcomps_1.html#K4">TopSpaceMetr</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="pre_topc.html#D3">PRE_TOPC:def 3</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"/><i><font color="Green">E6</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#K2">[#]</a> <font color="Maroon">c<sub>5</sub></font>
 
<b>by </b><i><a class="ref" href="pre_topc.html#D3">PRE_TOPC:def 3</a></i>;<br/>
<a NAME="E6:6_1_1"/><i><font color="Green">E7</font></i>: 
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>5</sub></font>
 
<b>by </b><i><a class="ref" href="vectmetr.html#D3">VECTMETR:def 3</a></i>;<br/>
<b>hence </b><i><font color="Green">E8</font></i>:  <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>6</sub></font> = 
the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="pcomps_1.html#K4">TopSpaceMetr</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="jordan24.html#E3:6_1_1"><i><font color="Green">E5</font></i></a></i>
<br/>.= 
 <a href="pre_topc.html#K2">[#]</a> <span class="p1">(<span class="default"><a href="pcomps_1.html#K4">TopSpaceMetr</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="pre_topc.html#D3">PRE_TOPC:def 3</a></i>
;<br/><br/>

<b>thus </b><a NAME="E8:6_1_1"/>
<font color="Maroon">c<sub>6</sub></font> is <a href="funct_1.html#V2">one-to-one</a>
 <b>by </b><i><a class="txt" href="jordan24.html#E3:6_1_1"><i><font color="Green">E5</font></i></a></i>;<br/>

<b>reconsider </b><font color="Maroon">c<sub>8</sub></font> = <font color="Maroon">c<sub>6</sub></font> as  <a href="jordan24.html#V1" target="_self">isometric</a> <a href="struct_0.html#NM7">Function</a> of <span class="p1">(<span class="default"><a href="pcomps_1.html#K4">TopSpaceMetr</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="pcomps_1.html#K4">TopSpaceMetr</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <b>by </b><i><a class="txt" href="jordan24.html#E1:6_1_1"><i><font color="Green">E4</font></i></a></i>;<br/>
<a NAME="E10:6_1_1"/>
<font color="Maroon">c<sub>8</sub></font> is <a href="pre_topc.html#V5">continuous</a>
 
;<br/>
<b>hence </b><a NAME="E11:6_1_1"/>
<font color="Maroon">c<sub>6</sub></font> is <a href="pre_topc.html#V5">continuous</a>
 ;<br/>

<a NAME="E12:6_1_1"/><i><font color="Green">E9</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="tops_2.html#NK3" target="_self">"</a>  is  <a href="vectmetr.html#V3">isometric</a> <a href="struct_0.html#NM7">Function</a> of <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>5</sub></font>
 
<b>by </b><i><a class="ref" href="vectmetr.html#T3">VECTMETR:3</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>9</sub></font> = <font color="Maroon">c<sub>7</sub></font> <a href="tops_2.html#NK3" target="_self">"</a>  as   <a href="struct_0.html#NM7">Function</a> of <span class="p1">(<span class="default"><a href="pcomps_1.html#K4">TopSpaceMetr</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="pcomps_1.html#K4">TopSpaceMetr</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> ;<br/>
<font color="Maroon">c<sub>6</sub></font> <a href="tops_2.html#NK3" target="_self">"</a>  = 
<font color="Maroon">c<sub>7</sub></font> <a href="funct_1.html#K2">"</a> 
<b>by </b><i><a class="txt" href="jordan24.html#E3:6_1_1"><i><font color="Green">E5</font></i></a>, <a class="ref" href="tops_2.html#D4">TOPS_2:def 4</a>, <a class="txt" href="jordan24.html#E7:6_1_1"><i><font color="Green">E8</font></i></a></i>
<br/>.= 
<font color="Maroon">c<sub>7</sub></font> <a href="tops_2.html#NK3" target="_self">"</a> 
<b>by </b><i><a class="txt" href="jordan24.html#E5:6_1_1"><i><font color="Green">E6</font></i></a>, <a class="txt" href="jordan24.html#E6:6_1_1"><i><font color="Green">E7</font></i></a>, <a class="ref" href="tops_2.html#D4">TOPS_2:def 4</a></i>
;<br/>
<a NAME="E15:6_1_1"/><b>then </b>
<font color="Maroon">c<sub>6</sub></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#NM7">Function</a> of <span class="p1">(<span class="default"><a href="pcomps_1.html#K4">TopSpaceMetr</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="pcomps_1.html#K4">TopSpaceMetr</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="jordan24.html#D2" target="_self">Def2</a>, <a class="txt" href="jordan24.html#E12:6_1_1"><i><font color="Green">E9</font></i></a></i>;<br/>
<b>hence </b><a NAME="E16:6_1_1"/>
<font color="Maroon">c<sub>6</sub></font> <a href="tops_2.html#NK3" target="_self">"</a>  is <a href="pre_topc.html#V5">continuous</a>
 ;<br/>


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