<?xml version="1.0"?>
<div class="add">

<b>let </b><font color="Maroon" title="c1">A</font> be  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>let </b><font color="Maroon" title="c2">f</font> be    <a href="fraenkel.html#M2" title="FRAENKEL:mode.2">Element</a> of  <a href="fraenkel.html#K1" title="FRAENKEL:func.1">Funcs</a> <font color="Maroon" title="c1">A</font>,<a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> ;<br/><b>let </b><font color="Maroon" title="c3">a</font>, <font color="Maroon" title="c4">b</font> be   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>;<br/>







<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="24_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c5">x</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">A</font>;<br/><b>thus </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="funcsdom.html#K8" title="FUNCSDOM:func.8">RealFuncExtMult</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="funcsdom.html#K2" title="FUNCSDOM:func.2">.</a> <span class="p2"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c3">a</font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="funcsdom.html#K8" title="FUNCSDOM:func.8">RealFuncExtMult</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="funcsdom.html#K2" title="FUNCSDOM:func.2">.</a> <span class="p4"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c4">b</font>,<font color="Maroon" title="c2">f</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span></span>)</span></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">x</font> = 
<font color="Maroon" title="c3">a</font> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="funcsdom.html#K8" title="FUNCSDOM:func.8">RealFuncExtMult</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="funcsdom.html#K2" title="FUNCSDOM:func.2">.</a> <span class="p3"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c4">b</font>,<font color="Maroon" title="c2">f</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">x</font></span>)</span>
<b>by </b><i><a class="ref" href="funcsdom.html#T15" target="_self" title="FUNCSDOM:th.15">Th15</a></i>
<br/>.= 
<font color="Maroon" title="c3">a</font> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">b</font> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">x</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="funcsdom.html#T15" target="_self" title="FUNCSDOM:th.15">Th15</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <font color="Maroon" title="c4">b</font></span>)</span> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">x</font></span>)</span>

<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="funcsdom.html#K8" title="FUNCSDOM:func.8">RealFuncExtMult</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="funcsdom.html#K2" title="FUNCSDOM:func.2">.</a> <span class="p2"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <font color="Maroon" title="c4">b</font></span>)</span>,<font color="Maroon" title="c2">f</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">x</font>
<b>by </b><i><a class="ref" href="funcsdom.html#T15" target="_self" title="FUNCSDOM:th.15">Th15</a></i>
;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E2:24"/>
<span class="p1">(<span class="default"><a href="funcsdom.html#K8" title="FUNCSDOM:func.8">RealFuncExtMult</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="funcsdom.html#K2" title="FUNCSDOM:func.2">.</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>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="funcsdom.html#K8" title="FUNCSDOM:func.8">RealFuncExtMult</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="funcsdom.html#K2" title="FUNCSDOM:func.2">.</a> <span class="p3"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c4">b</font>,<font color="Maroon" title="c2">f</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span></span>)</span></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <span class="p1">(<span class="default"><a href="funcsdom.html#K8" title="FUNCSDOM:func.8">RealFuncExtMult</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="funcsdom.html#K2" title="FUNCSDOM:func.2">.</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <font color="Maroon" title="c4">b</font></span>)</span>,<font color="Maroon" title="c2">f</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>
 <b>by </b><i><a class="ref" href="funct_2.html#T113" title="FUNCT_2:th.113">FUNCT_2:113</a></i>;<br/>


</div>
