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<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> ,   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="rlvect_2.html#K11" title="RLVECT_2:func.11">LinComb</a> <font color="Maroon" title="c1">V</font>,   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <b>means </b> ex <font color="Olive" title="b1">a</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> st <br/>( <font color="Olive" title="b1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> $1 &amp; $3 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">a</font> <a href="rlvect_2.html#K8" title="RLVECT_2:func.8">*</a> <span class="p1">(<span class="default"><a href="rlvect_2.html#K12" title="RLVECT_2:func.12">@</a> $2</span>)</span> );<br/>
<a NAME="E1:76_1_1"/><i><font color="Green" title="E45">A1</font></i>: 
for <font color="Olive" title="b1">x</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> <br/> for <font color="Olive" title="b2">e1</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="rlvect_2.html#K11" title="RLVECT_2:func.11">LinComb</a> <font color="Maroon" title="c1">V</font>  ex <font color="Olive" title="b3">e2</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="rlvect_2.html#K11" title="RLVECT_2:func.11">LinComb</a> <font color="Maroon" title="c1">V</font> st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">e1</font>,<font color="Olive" title="b3">e2</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="76_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c2">x</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> ;<br/><b>let </b><font color="Maroon" title="c3">e1</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="rlvect_2.html#K11" title="RLVECT_2:func.11">LinComb</a> <font color="Maroon" title="c1">V</font>;<br/>



<b>take </b>
 <a href="rlvect_2.html#K13" title="RLVECT_2:func.13">@</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="rlvect_2.html#K8" title="RLVECT_2:func.8">*</a> <span class="p2">(<span class="default"><a href="rlvect_2.html#K12" title="RLVECT_2:func.12">@</a> <font color="Maroon" title="c3">e1</font></span>)</span></span>)</span>
;<br/>

<b>take </b>
<font color="Maroon" title="c2">x</font>
;<br/>

<b>thus </b><a NAME="E1:76_1_1_1"/>
( <font color="Maroon" title="c2">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">x</font> &amp;  <a href="rlvect_2.html#K13" title="RLVECT_2:func.13">@</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="rlvect_2.html#K8" title="RLVECT_2:func.8">*</a> <span class="p2">(<span class="default"><a href="rlvect_2.html#K12" title="RLVECT_2:func.12">@</a> <font color="Maroon" title="c3">e1</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">x</font> <a href="rlvect_2.html#K8" title="RLVECT_2:func.8">*</a> <span class="p1">(<span class="default"><a href="rlvect_2.html#K12" title="RLVECT_2:func.12">@</a> <font color="Maroon" title="c3">e1</font></span>)</span> )
 ;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon" title="c2">g</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> ,<span class="p2">(<span class="default"><a href="rlvect_2.html#K11" title="RLVECT_2:func.11">LinComb</a> <font color="Maroon" title="c1">V</font></span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>, <a href="rlvect_2.html#K11" title="RLVECT_2:func.11">LinComb</a> <font color="Maroon" title="c1">V</font><b> such that </b><br/><a NAME="E3:76_1_1"/><i><font color="Green" title="E46">A2</font></i>: 
for <font color="Olive" title="b1">x</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> <br/> for <font color="Olive" title="b2">e</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="rlvect_2.html#K11" title="RLVECT_2:func.11">LinComb</a> <font color="Maroon" title="c1">V</font> holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">e</font>,<font color="Maroon" title="c2">g</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">e</font>]
 <b>from </b><i><a class="ref" href="binop_1.html#S3" title="BINOP_1:sch.3">BINOP_1:sch 3</a>(<a class="txt" href="rlvect_2.html#E1:76_1_1"><i><font color="Green" title="E45">A1</font></i></a>);<br/></i>
<b>take </b>
<font color="Maroon" title="c2">g</font>
;<br/>

<b>let </b><font color="Maroon" title="c3">a</font> be   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>;<br/><b>let </b><font color="Maroon" title="c4">e</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="rlvect_2.html#K11" title="RLVECT_2:func.11">LinComb</a> <font color="Maroon" title="c1">V</font>;<br/>



<a NAME="E4:76_1_1"/>
 ex <font color="Olive" title="b1">b</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> st <br/>( <font color="Olive" title="b1">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">a</font> &amp; <font color="Maroon" title="c2">g</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">b</font> <a href="rlvect_2.html#K8" title="RLVECT_2:func.8">*</a> <span class="p1">(<span class="default"><a href="rlvect_2.html#K12" title="RLVECT_2:func.12">@</a> <font color="Maroon" title="c4">e</font></span>)</span> )
 
<b>by </b><i><a class="txt" href="rlvect_2.html#E3:76_1_1"><i><font color="Green" title="E46">A2</font></i></a></i>;<br/>
<b>hence </b><a NAME="E5:76_1_1"/>
<font color="Maroon" title="c2">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">e</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">a</font> <a href="rlvect_2.html#K8" title="RLVECT_2:func.8">*</a> <span class="p1">(<span class="default"><a href="rlvect_2.html#K12" title="RLVECT_2:func.12">@</a> <font color="Maroon" title="c4">e</font></span>)</span>
 ;<br/>


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