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<b>let </b><font color="Maroon" title="c1">X</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">a</font> be   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>;<br/><b>let </b><font color="Maroon" title="c3">F</font>, <font color="Maroon" title="c4">G</font> be   <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <span class="p1">(<span class="default"><a href="c0sp1.html#K7" title="C0SP1:func.7">R_Algebra_of_BoundedFunctions</a> <font color="Maroon" title="c1">X</font></span>)</span>;<br/><b>let </b><font color="Maroon" title="c5">f</font>, <font color="Maroon" title="c6">g</font> be   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c1">X</font>, <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> ;<br/>











<b>assume </b><a NAME="E1:37"/><i><font color="Green" title="E25">A1</font></i>: 
( <font color="Maroon" title="c5">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">F</font> &amp; <font color="Maroon" title="c6">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">G</font> )
 ;<br/>

<a NAME="E2:37"/><i><font color="Green" title="E26">A3</font></i>: 
 <a href="c0sp1.html#K7" title="C0SP1:func.7">R_Algebra_of_BoundedFunctions</a> <font color="Maroon" title="c1">X</font> is    <a href="c0sp1.html#M2" title="C0SP1:mode.2">Subalgebra</a> of  <a href="funcsdom.html#K13" title="FUNCSDOM:func.13">RAlgebra</a> <font color="Maroon" title="c1">X</font>
 
<b>by </b><i><a class="ref" href="c0sp1.html#T6" target="_self" title="C0SP1:th.6">Th03</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c7">f1</font> = <font color="Maroon" title="c3">F</font>, <font color="Maroon" title="c8">g1</font> = <font color="Maroon" title="c4">G</font> as   <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <span class="p1">(<span class="default"><a href="funcsdom.html#K13" title="FUNCSDOM:func.13">RAlgebra</a> <font color="Maroon" title="c1">X</font></span>)</span> <b>by </b><i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></i>;<br/>
<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:37_1"/><i><font color="Green" title="E27">A5</font></i>: 
<font color="Maroon" title="c4">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c3">F</font>
 ;<br/><b>let </b><font color="Maroon" title="c9">x</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">X</font>;<br/><a NAME="E2:37_1"/>
<font color="Maroon" title="c8">g1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c7">f1</font>
 <b>by </b><i><a class="txt" href="c0sp1.html#E2:37"><i><font color="Green" title="E26">A3</font></i></a>, <a class="txt" href="c0sp1.html#E1:37_1"><i><font color="Green" title="E27">A5</font></i></a>, <a class="ref" href="c0sp1.html#T8" target="_self" title="C0SP1:th.8">RLSUB121</a></i>;<br/><b>hence </b><a NAME="E3:37_1"/>
<font color="Maroon" title="c6">g</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c9">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">a</font> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c9">x</font></span>)</span>
 <b>by </b><i><a class="txt" href="c0sp1.html#E1:37"><i><font color="Green" title="E25">A1</font></i></a>, <a class="ref" href="funcsdom.html#T15" title="FUNCSDOM:th.15">FUNCSDOM:15</a></i>;<br/></div>
<b>end;</b></div>

<b>assume </b><a NAME="E5:37"/>
for <font color="Olive" title="b1">x</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">X</font> holds  <font color="Maroon" title="c6">g</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">a</font> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">x</font></span>)</span>
 ;<br/>

<a NAME="E6:37"/><b>then </b>
<font color="Maroon" title="c8">g1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c7">f1</font>
 
<b>by </b><i><a class="txt" href="c0sp1.html#E1:37"><i><font color="Green" title="E25">A1</font></i></a>, <a class="ref" href="funcsdom.html#T15" title="FUNCSDOM:th.15">FUNCSDOM:15</a></i>;<br/>
<b>hence </b><a NAME="E7:37"/>
<font color="Maroon" title="c4">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c3">F</font>
 <b>by </b><i><a class="txt" href="c0sp1.html#E2:37"><i><font color="Green" title="E26">A3</font></i></a>, <a class="ref" href="c0sp1.html#T8" target="_self" title="C0SP1:th.8">RLSUB121</a></i>;<br/>


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