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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> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">F</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="c0sp1.html#K11" title="C0SP1:func.11">R_Normed_Algebra_of_BoundedFunctions</a> <font color="Maroon" title="c1">X</font></span>)</span> holds  <span class="p1">(<span class="default"><a href="c0sp1.html#K5" title="C0SP1:func.5">Mult_</a> <span class="p2">(<span class="default"><a href="c0sp1.html#K6" title="C0SP1:func.6">BoundedFunctions</a> <font color="Maroon" title="c1">X</font></span>)</span>,<span class="p2">(<span class="default"><a href="funcsdom.html#K13" title="FUNCSDOM:func.13">RAlgebra</a> <font color="Maroon" title="c1">X</font></span>)</span></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> 1,<font color="Olive" title="b1">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">F</font></span><br/><b>let </b><font color="Maroon" title="c2">F</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="c0sp1.html#K11" title="C0SP1:func.11">R_Normed_Algebra_of_BoundedFunctions</a> <font color="Maroon" title="c1">X</font></span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><a href="c0sp1.html#K5" title="C0SP1:func.5">Mult_</a> <span class="p2">(<span class="default"><a href="c0sp1.html#K6" title="C0SP1:func.6">BoundedFunctions</a> <font color="Maroon" title="c1">X</font></span>)</span>,<span class="p2">(<span class="default"><a href="funcsdom.html#K13" title="FUNCSDOM:func.13">RAlgebra</a> <font color="Maroon" title="c1">X</font></span>)</span></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> 1,<font color="Maroon" title="c2">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">F</font></span><br/>



<b>set </b><font color="Maroon" title="c3">X1</font> =  <a href="c0sp1.html#K6" title="C0SP1:func.6">BoundedFunctions</a> <font color="Maroon" title="c1">X</font>;<br/>
<b>reconsider </b><font color="Maroon" title="c4">f1</font> = <font color="Maroon" title="c2">F</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="c0sp1.html#K6" title="C0SP1:func.6">BoundedFunctions</a> <font color="Maroon" title="c1">X</font> ;<br/>
<a NAME="E2:55"/><i><font color="Green" title="E38">A2</font></i>: 
<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default">1,<font color="Maroon" title="c4">f1</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <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="c0sp1.html#K6" title="C0SP1:func.6">BoundedFunctions</a> <font color="Maroon" title="c1">X</font></span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></i>;<br/>
<b>thus </b><span class="p1">(<span class="default"><a href="c0sp1.html#K5" title="C0SP1:func.5">Mult_</a> <span class="p2">(<span class="default"><a href="c0sp1.html#K6" title="C0SP1:func.6">BoundedFunctions</a> <font color="Maroon" title="c1">X</font></span>)</span>,<span class="p2">(<span class="default"><a href="funcsdom.html#K13" title="FUNCSDOM:func.13">RAlgebra</a> <font color="Maroon" title="c1">X</font></span>)</span></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> 1,<font color="Maroon" title="c2">F</font> = 
<span class="p1">(<span class="default">the <a href="rlvect_1.html#U1" title="RLVECT_1:sel.1">Mult</a> of <span class="p2">(<span class="default"><a href="funcsdom.html#K13" title="FUNCSDOM:func.13">RAlgebra</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2"><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="p3">(<span class="default"><a href="c0sp1.html#K6" title="C0SP1:func.6">BoundedFunctions</a> <font color="Maroon" title="c1">X</font></span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> 1,<font color="Maroon" title="c4">f1</font>
<b>by </b><i><a class="ref" href="c0sp1.html#D11" target="_self" title="C0SP1:def.11">def212</a></i>
<br/>.= 
the <a href="rlvect_1.html#U1" title="RLVECT_1:sel.1">Mult</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> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> 1,<font color="Maroon" title="c4">f1</font>
<b>by </b><i><a class="txt" href="c0sp1.html#E2:55"><i><font color="Green" title="E38">A2</font></i></a>, <a class="ref" href="funct_1.html#T72" title="FUNCT_1:th.72">FUNCT_1:72</a></i>
<br/>.= 
<font color="Maroon" title="c2">F</font>
<b>by </b><i><a class="ref" href="funcsdom.html#T23" title="FUNCSDOM:th.23">FUNCSDOM:23</a></i>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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