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<span class="kw">let </span><font color="Maroon" title="c2">X</font> be   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a> ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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="c2">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="c2">X</font></span>)</span>,<span class="p2">(<span class="default"><a href="funcsdom.html#K12" title="FUNCSDOM:func.12">RAlgebra</a> <font color="Maroon" title="c2">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/><span class="kw">let </span><font color="Maroon" title="c3">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="c2">X</font></span>)</span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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="c2">X</font></span>)</span>,<span class="p2">(<span class="default"><a href="funcsdom.html#K12" title="FUNCSDOM:func.12">RAlgebra</a> <font color="Maroon" title="c2">X</font></span>)</span>)</span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (1,<font color="Maroon" title="c3">F</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">F</font></span><br/>



<span class="kw">set </span><font color="Maroon" title="c4">X1</font> =  <a href="c0sp1.html#K6" title="C0SP1:func.6">BoundedFunctions</a> <font color="Maroon" title="c2">X</font>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c5">f1</font> = <font color="Maroon" title="c3">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="c2">X</font> ;<br/>
<a NAME="E2:52"/><span class="lab"><font color="Green" title="E31">A1</font></span>: 
<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c1">jj</font>,<font color="Maroon" title="c5">f1</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="tarski.html#R2" title="TARSKI: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="c2">X</font></span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 
;<br/>
<span class="kw">thus </span><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="c2">X</font></span>)</span>,<span class="p2">(<span class="default"><a href="funcsdom.html#K12" title="FUNCSDOM:func.12">RAlgebra</a> <font color="Maroon" title="c2">X</font></span>)</span>)</span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (1,<font color="Maroon" title="c3">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#K12" title="FUNCSDOM:func.12">RAlgebra</a> <font color="Maroon" title="c2">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="c2">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="c5">f1</font>)
<span class="kw">by</span> <span class="lab"><a class="ref" href="c0sp1.html#D11" target="_self" onmouseover="rs('c0sp1/D11')" onmouseout="rh()">Def11</a></span>
<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#K12" title="FUNCSDOM:func.12">RAlgebra</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (1,<font color="Maroon" title="c5">f1</font>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:52"><span class="lab"><font color="Green" title="E31">A1</font></span></a>, <a class="ref" href="funct_1.html#T49" onmouseover="rs('funct_1/T49')" onmouseout="rh()">FUNCT_1:49</a></span>
<br/>.= 
<font color="Maroon" title="c3">F</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="funcsdom.html#T12" onmouseover="rs('funcsdom/T12')" onmouseout="rh()">FUNCSDOM:12</a></span>
; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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