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<b>let </b><font color="Maroon">c<sub>1</sub></font> be   <a href="struct_0.html#NM7">Function</a> of <a href="topmetr.html#K3">R^1</a> ,<a href="topmetr.html#K3">R^1</a> ;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font>, <font color="Maroon">c<sub>3</sub></font> be   <a href="real_1.html#NM1">Real</a>;<br/>



<b>assume </b><b>that </b><br/><a NAME="E1:39"/><i><font color="Green">E24</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 <b>and </b><br/><a NAME="E2:39"/><i><font color="Green">E25</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="funct_2.html#R1">=</a>  <a href="jordan16.html#K1" target="_self">AffineMap</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>
 ;<br/>

<a NAME="E3:39"/><i><font color="Green">E26</font></i>: 
 <a href="pre_topc.html#K2">[#]</a> <a href="topmetr.html#K3">R^1</a>  <a href="hidden.html#R1">=</a>  <a href="numbers.html#K1">REAL</a> 
 
<b>by </b><i><a class="ref" href="pre_topc.html#T12">PRE_TOPC:12</a>, <a class="ref" href="topmetr.html#T24">TOPMETR:24</a></i>;<br/>
<b>hence </b><a NAME="E4:39"/>
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#K2">[#]</a> <a href="topmetr.html#K3">R^1</a> 
 <b>by </b><i><a class="txt" href="jordan16.html#E2:39"><i><font color="Green">E25</font></i></a>, <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/>

<b>thus </b><a NAME="E5:39"/><i><font color="Green">E27</font></i>: 
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#K2">[#]</a> <a href="topmetr.html#K3">R^1</a> 
 <b>by </b><i><a class="txt" href="jordan16.html#E1:39"><i><font color="Green">E24</font></i></a>, <a class="txt" href="jordan16.html#E2:39"><i><font color="Green">E25</font></i></a>, <a class="txt" href="jordan16.html#E3:39"><i><font color="Green">E26</font></i></a>, <a class="ref" href="jordan16.html#T32" target="_self">Th32</a></i>;<br/>

<b>thus </b><a NAME="E6:39"/><i><font color="Green">E28</font></i>: 
<font color="Maroon">c<sub>1</sub></font> is <a href="funct_1.html#V2">one-to-one</a>
 <b>by </b><i><a class="txt" href="jordan16.html#E1:39"><i><font color="Green">E24</font></i></a>, <a class="txt" href="jordan16.html#E2:39"><i><font color="Green">E25</font></i></a>, <a class="ref" href="jordan16.html#T27" target="_self">Th27</a></i>;<br/>

<a NAME="E7:39"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="real_1.html#NM1">Real</a> holds <font color="Maroon">c<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="real_1.html#K4">*</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="real_1.html#K3">+</a> <font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="txt" href="jordan16.html#E2:39"><i><font color="Green">E25</font></i></a>, <a class="ref" href="jordan16.html#D3" target="_self">Def3</a></i>;<br/>
<b>hence </b><a NAME="E8:39"/>
<font color="Maroon">c<sub>1</sub></font> is <a href="pre_topc.html#V5">continuous</a>
 <b>by </b><i><a class="ref" href="topmetr.html#T28">TOPMETR:28</a></i>;<br/>

<font color="Maroon">c<sub>1</sub></font> <a href="tops_2.html#NK3" target="_self">"</a>  = 
<font color="Maroon">c<sub>1</sub></font> <a href="funct_1.html#K2">"</a> 
<b>by </b><i><a class="txt" href="jordan16.html#E5:39"><i><font color="Green">E27</font></i></a>, <a class="txt" href="jordan16.html#E6:39"><i><font color="Green">E28</font></i></a>, <a class="ref" href="tops_2.html#D4">TOPS_2:def 4</a></i>
<br/>.= 
 <a href="jordan16.html#K1" target="_self">AffineMap</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="real_1.html#K2">"</a> </span>)</span>,<span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="real_1.html#K6">/</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="jordan16.html#E1:39"><i><font color="Green">E24</font></i></a>, <a class="txt" href="jordan16.html#E2:39"><i><font color="Green">E25</font></i></a>, <a class="ref" href="jordan16.html#T33" target="_self">Th33</a></i>
;<br/>
<a NAME="E10:39"/><b>then </b>
for <font color="Olive">b<sub>1</sub></font> being  <a href="real_1.html#NM1">Real</a> holds <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="tops_2.html#NK3" target="_self">"</a> </span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="real_1.html#K2">"</a> </span>)</span> <a href="real_1.html#K4">*</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="real_1.html#K6">/</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="jordan16.html#D3" target="_self">Def3</a></i>;<br/>
<b>hence </b><a NAME="E11:39"/>
<font color="Maroon">c<sub>1</sub></font> <a href="tops_2.html#NK3" target="_self">"</a>  is <a href="pre_topc.html#V5">continuous</a>
 <b>by </b><i><a class="ref" href="topmetr.html#T28">TOPMETR:28</a></i>;<br/>


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