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<a NAME="E1:22_1_1"/><i><font color="Green">E22</font></i>: 
<span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><a href="numbers.html#K1">REAL</a> ,<font color="Maroon">c<sub>2</sub></font></span><a href="zfmisc_1.html#K2">:]</a></span> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><a href="numbers.html#K1">REAL</a> ,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font></span><a href="zfmisc_1.html#K2">:]</a></span>
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T118">ZFMISC_1:118</a></i>;<br/>
<a NAME="E2:22_1_1"/><i><font color="Green">E23</font></i>: 
 <a href="relat_1.html#K1">dom</a> the <a href="rlvect_1.html#U2">Mult</a> of <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><a href="numbers.html#K1">REAL</a> ,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font></span><a href="zfmisc_1.html#K2">:]</a></span>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E3:22_1_1"/><b>then </b><i><font color="Green">E24</font></i>: 
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default">the <a href="rlvect_1.html#U2">Mult</a> of <font color="Maroon">c<sub>1</sub></font> <a href="partfun1.html#K2">|</a> <span class="p2"><a href="zfmisc_1.html#K2">[:</a><span class="default"><a href="numbers.html#K1">REAL</a> ,<font color="Maroon">c<sub>2</sub></font></span><a href="zfmisc_1.html#K2">:]</a></span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><a href="numbers.html#K1">REAL</a> ,<font color="Maroon">c<sub>2</sub></font></span><a href="zfmisc_1.html#K2">:]</a></span>
 
<b>by </b><i><a class="txt" href="rsspace.html#E1:22_1_1"><i><font color="Green">E22</font></i></a>, <a class="ref" href="relat_1.html#T91">RELAT_1:91</a></i>;<br/>
<a NAME="E4:22_1_1"/>
for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><a href="numbers.html#K1">REAL</a> ,<font color="Maroon">c<sub>2</sub></font></span><a href="zfmisc_1.html#K2">:]</a></span> holds <br/><span class="p1">(<span class="default">the <a href="rlvect_1.html#U2">Mult</a> of <font color="Maroon">c<sub>1</sub></font> <a href="partfun1.html#K2">|</a> <span class="p2"><a href="zfmisc_1.html#K2">[:</a><span class="default"><a href="numbers.html#K1">REAL</a> ,<font color="Maroon">c<sub>2</sub></font></span><a href="zfmisc_1.html#K2">:]</a></span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>3</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:22_1_1_1"/><i><font color="Green">E25</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><a href="numbers.html#K1">REAL</a> ,<font color="Maroon">c<sub>2</sub></font></span><a href="zfmisc_1.html#K2">:]</a></span>
 ;<br/>

<b>consider </b><font color="Maroon">c<sub>4</sub></font>, <font color="Maroon">c<sub>5</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:22_1_1_1"/><i><font color="Green">E26</font></i>: 
( <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K1">REAL</a>  &amp; <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font></span><a href="tarski.html#K4">]</a></span> )
 <b>by </b><i><a class="txt" href="rsspace.html#E1:22_1_1_1"><i><font color="Green">E25</font></i></a>, <a class="ref" href="zfmisc_1.html#D2">ZFMISC_1:def 2</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>6</sub></font> = <font color="Maroon">c<sub>5</sub></font> as   <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="txt" href="rsspace.html#E3:22_1_1_1"><i><font color="Green">E26</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>7</sub></font> = <font color="Maroon">c<sub>4</sub></font> as   <a href="real_1.html#NM1">Real</a> <b>by </b><i><a class="txt" href="rsspace.html#E3:22_1_1_1"><i><font color="Green">E26</font></i></a></i>;<br/>
<a NAME="E6:22_1_1_1"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>5</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default">the <a href="rlvect_1.html#U2">Mult</a> of <font color="Maroon">c<sub>1</sub></font> <a href="partfun1.html#K2">|</a> <span class="p2"><a href="zfmisc_1.html#K2">[:</a><span class="default"><a href="numbers.html#K1">REAL</a> ,<font color="Maroon">c<sub>2</sub></font></span><a href="zfmisc_1.html#K2">:]</a></span></span>)</span>
 
<b>by </b><i><a class="txt" href="rsspace.html#E1:22_1_1"><i><font color="Green">E22</font></i></a>, <a class="txt" href="rsspace.html#E2:22_1_1"><i><font color="Green">E23</font></i></a>, <a class="txt" href="rsspace.html#E1:22_1_1_1"><i><font color="Green">E25</font></i></a>, <a class="txt" href="rsspace.html#E3:22_1_1_1"><i><font color="Green">E26</font></i></a>, <a class="ref" href="relat_1.html#T91">RELAT_1:91</a></i>;<br/>
<b>then </b><span class="p1">(<span class="default">the <a href="rlvect_1.html#U2">Mult</a> of <font color="Maroon">c<sub>1</sub></font> <a href="partfun1.html#K2">|</a> <span class="p2"><a href="zfmisc_1.html#K2">[:</a><span class="default"><a href="numbers.html#K1">REAL</a> ,<font color="Maroon">c<sub>2</sub></font></span><a href="zfmisc_1.html#K2">:]</a></span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>3</sub></font> = 
the <a href="rlvect_1.html#U2">Mult</a> of <font color="Maroon">c<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>5</sub></font></span><a href="tarski.html#K4">]</a></span>
<b>by </b><i><a class="txt" href="rsspace.html#E3:22_1_1_1"><i><font color="Green">E26</font></i></a>, <a class="ref" href="funct_1.html#T70">FUNCT_1:70</a></i>
<br/>.= 
<font color="Maroon">c<sub>7</sub></font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">c<sub>6</sub></font>

;<br/>
<b>hence </b><a NAME="E8:22_1_1_1"/>
<span class="p1">(<span class="default">the <a href="rlvect_1.html#U2">Mult</a> of <font color="Maroon">c<sub>1</sub></font> <a href="partfun1.html#K2">|</a> <span class="p2"><a href="zfmisc_1.html#K2">[:</a><span class="default"><a href="numbers.html#K1">REAL</a> ,<font color="Maroon">c<sub>2</sub></font></span><a href="zfmisc_1.html#K2">:]</a></span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="rsspace.html#E1:22"><i><font color="Green">E21</font></i></a>, <a class="txt" href="rsspace.html#E3:22_1_1_1"><i><font color="Green">E26</font></i></a>, <a class="ref" href="rlsub_1.html#D1">RLSUB_1:def 1</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E5:22_1_1"/>
the <a href="rlvect_1.html#U2">Mult</a> of <font color="Maroon">c<sub>1</sub></font> <a href="partfun1.html#K2">|</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><a href="numbers.html#K1">REAL</a> ,<font color="Maroon">c<sub>2</sub></font></span><a href="zfmisc_1.html#K2">:]</a></span> is   <a href="funct_2.html#NM1">Function</a> of <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><a href="numbers.html#K1">REAL</a> ,<font color="Maroon">c<sub>2</sub></font></span><a href="zfmisc_1.html#K2">:]</a></span>,<font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="rsspace.html#E3:22_1_1"><i><font color="Green">E24</font></i></a>, <a class="ref" href="funct_2.html#T5">FUNCT_2:5</a></i>;<br/>


</div>
