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<b>let </b><font color="Maroon">mult1</font> be   <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> ,<span class="p2">(<span class="default"><a href="euclid.html#K1">REAL</a> <font color="Maroon">n</font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span>, <a href="euclid.html#K1">REAL</a> <font color="Maroon">n</font>, <font color="Maroon">mult2</font> be   <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> ,<span class="p2">(<span class="default"><a href="euclid.html#K1">REAL</a> <font color="Maroon">n</font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span>, <a href="euclid.html#K1">REAL</a> <font color="Maroon">n</font>;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:5_1_1"/><i><font color="Green">E33</font></i>: 
for <font color="Olive">r</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a> <br/> for <font color="Olive">x</font> being   <a href="finseq_2.html#M2">Element</a> of  <a href="euclid.html#K1">REAL</a> <font color="Maroon">n</font> holds  <font color="Maroon">mult1</font> <a href="binop_1.html#K2">.</a> <font color="Olive">r</font>,<font color="Olive">x</font> <a href="hidden.html#R1">=</a> <font color="Olive">r</font> <a href="euclid.html#K9">*</a> <font color="Olive">x</font>
 <b>and </b><br/><a NAME="E2:5_1_1"/><i><font color="Green">E34</font></i>: 
for <font color="Olive">r</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a> <br/> for <font color="Olive">x</font> being   <a href="finseq_2.html#M2">Element</a> of  <a href="euclid.html#K1">REAL</a> <font color="Maroon">n</font> holds  <font color="Maroon">mult2</font> <a href="binop_1.html#K2">.</a> <font color="Olive">r</font>,<font color="Olive">x</font> <a href="hidden.html#R1">=</a> <font color="Olive">r</font> <a href="euclid.html#K9">*</a> <font color="Olive">x</font>
 ;<br/>

<a NAME="E3:5_1_1"/>
for <font color="Olive">r</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a> <br/> for <font color="Olive">x</font> being   <a href="finseq_2.html#M2">Element</a> of  <a href="euclid.html#K1">REAL</a> <font color="Maroon">n</font> holds  <font color="Maroon">mult1</font> <a href="binop_1.html#K2">.</a> <font color="Olive">r</font>,<font color="Olive">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">mult2</font> <a href="binop_1.html#K2">.</a> <font color="Olive">r</font>,<font color="Olive">x</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="5_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">r</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a> ;<br/><b>let </b><font color="Maroon">x</font> be    <a href="finseq_2.html#M2">Element</a> of  <a href="euclid.html#K1">REAL</a> <font color="Maroon">n</font>;<br/>



<b>thus </b><font color="Maroon">mult1</font> <a href="binop_1.html#K2">.</a> <font color="Maroon">r</font>,<font color="Maroon">x</font> = 
<font color="Maroon">r</font> <a href="euclid.html#K9">*</a> <font color="Maroon">x</font>
<b>by </b><i/>
<br/>.= 
<font color="Maroon">mult2</font> <a href="binop_1.html#K2">.</a> <font color="Maroon">r</font>,<font color="Maroon">x</font>
<b>by </b><i/>
;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:5_1_1"/>
<font color="Maroon">mult1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">mult2</font>
 <b>by </b><i><a class="ref" href="binop_1.html#T2">BINOP_1:2</a></i>;<br/>


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