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<b>let </b><font color="Maroon">A</font> be  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">h</font> be    <a href="fraenkel.html#M2">Element</a> of  <a href="fraenkel.html#K1">Funcs</a> <font color="Maroon">A</font>,<a href="numbers.html#K1">REAL</a> ;<br/><b>let </b><font color="Maroon">f</font> be    <a href="fraenkel.html#M2">Element</a> of  <a href="fraenkel.html#K1">Funcs</a> <font color="Maroon">A</font>,<a href="numbers.html#K1">REAL</a> ;<br/><b>let </b><font color="Maroon">g</font> be    <a href="fraenkel.html#M2">Element</a> of  <a href="fraenkel.html#K1">Funcs</a> <font color="Maroon">A</font>,<a href="numbers.html#K1">REAL</a> ;<br/>







<div><i><font color="Green">E36</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="13_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:13_1"/><i><font color="Green">E41</font></i>: 
<font color="Maroon">h</font> <a href="funct_2.html#R2">=</a> <span class="p1">(<span class="default"><a href="funcsdom.html#K7" target="_self">RealFuncMult</a> <font color="Maroon">A</font></span>)</span> <a href="funcsdom.html#K1" target="_self">.</a> <font color="Maroon">f</font>,<font color="Maroon">g</font>
 ;<br/><b>let </b><font color="Maroon">x</font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">A</font>;<br/><a NAME="E2:13_1"/><i><font color="Green">E42</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="binop_2.html#K35">multreal</a>  <a href="funcsdom.html#K4" target="_self">.:</a> <font color="Maroon">f</font>,<font color="Maroon">g</font></span>)</span>
 <b>by </b><i/>;<br/><b>thus </b><font color="Maroon">h</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font> = 
<span class="p1">(<span class="default"><a href="binop_2.html#K35">multreal</a>  <a href="funcsdom.html#K4" target="_self">.:</a> <font color="Maroon">f</font>,<font color="Maroon">g</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font>
<b>by </b><i>, </i>
<br/>.= 
<a href="binop_2.html#K35">multreal</a>  <a href="binop_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span>
<b>by </b><i>, <a class="ref" href="funcop_1.html#T28">FUNCOP_1:28</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span>
<b>by </b><i><a class="ref" href="binop_2.html#D11">BINOP_2:def 11</a></i>
;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="13_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:13_2"/><i><font color="Green">E47</font></i>: 
for <font color="Olive">x</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">A</font> holds  <font color="Maroon">h</font> <a href="funct_2.html#K3">.</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Olive">x</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Olive">x</font></span>)</span>
 ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="13_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">x</font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">A</font>;<br/><a NAME="E1:13_2_1"/><i><font color="Green">E48</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="binop_2.html#K35">multreal</a>  <a href="funcsdom.html#K4" target="_self">.:</a> <font color="Maroon">f</font>,<font color="Maroon">g</font></span>)</span>
 <b>by </b><i/>;<br/><b>thus </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="funcsdom.html#K7" target="_self">RealFuncMult</a> <font color="Maroon">A</font></span>)</span> <a href="funcsdom.html#K1" target="_self">.</a> <font color="Maroon">f</font>,<font color="Maroon">g</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font> = 
<span class="p1">(<span class="default"><a href="binop_2.html#K35">multreal</a>  <a href="funcsdom.html#K4" target="_self">.:</a> <font color="Maroon">f</font>,<font color="Maroon">g</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font>
<b>by </b><i/>
<br/>.= 
<a href="binop_2.html#K35">multreal</a>  <a href="binop_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span>
<b>by </b><i>, <a class="ref" href="funcop_1.html#T28">FUNCOP_1:28</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span>
<b>by </b><i><a class="ref" href="binop_2.html#D11">BINOP_2:def 11</a></i>
<br/>.= 
<font color="Maroon">h</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font>
<b>by </b><i/>
;<br/></div><b>end;</b></div><b>hence </b><a NAME="E3:13_2"/>
<font color="Maroon">h</font> <a href="funct_2.html#R2">=</a> <span class="p1">(<span class="default"><a href="funcsdom.html#K7" target="_self">RealFuncMult</a> <font color="Maroon">A</font></span>)</span> <a href="funcsdom.html#K1" target="_self">.</a> <font color="Maroon">f</font>,<font color="Maroon">g</font>
 <b>by </b><i><a class="ref" href="funct_2.html#T113">FUNCT_2:113</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E3:13"/>
( <font color="Maroon">h</font> <a href="funct_2.html#R2">=</a> <span class="p1">(<span class="default"><a href="funcsdom.html#K7" target="_self">RealFuncMult</a> <font color="Maroon">A</font></span>)</span> <a href="funcsdom.html#K1" target="_self">.</a> <font color="Maroon">f</font>,<font color="Maroon">g</font> iff for <font color="Olive">x</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">A</font> holds  <font color="Maroon">h</font> <a href="funct_2.html#K3">.</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Olive">x</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Olive">x</font></span>)</span> )
 <b>by </b><i/>;<br/>


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