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<span class="kw">let </span><font color="Maroon" title="c1">X</font> be   non  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>   <a href="compts_1.html#V1" title="COMPTS_1:attr.1">compact</a>  <a href="pre_topc.html#NM1" title="PRE_TOPC:NM.1">TopSpace</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="cc0sp2.html#K3" title="CC0SP2:func.3">C_Algebra_of_ContinuousFunctions</a> <font color="Maroon" title="c1">X</font> is    <a href="cc0sp1.html#M1" title="CC0SP1:mode.1">ComplexSubAlgebra</a> of  <a href="cc0sp1.html#K3" title="CC0SP1:func.3">C_Algebra_of_BoundedFunctions</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">X</font></span><br/>

<a NAME="E1:26"/><span class="lab"><font color="Green" title="E16">A1</font></span>: 
 the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="cc0sp2.html#K3" title="CC0SP2:func.3">C_Algebra_of_ContinuousFunctions</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="cc0sp1.html#K3" title="CC0SP1:func.3">C_Algebra_of_BoundedFunctions</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">X</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="26_1"><span class="kw">proof </span></a><div class="add">

<a NAME="E1:26_1"/>
 for <font color="Olive" title="b1">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cc0sp2.html#K2" title="CC0SP2:func.2">CContinuousFunctions</a> <font color="Maroon" title="c1">X</font> holds <br/><font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cc0sp1.html#K2" title="CC0SP1:func.2">ComplexBoundedFunctions</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">X</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="cc0sp2.html#E20"><span class="lab"><font color="Green" title="E15">Lm1</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E2:26_1"/>
 the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="cc0sp2.html#K3" title="CC0SP2:func.3">C_Algebra_of_ContinuousFunctions</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="cc0sp1.html#K3" title="CC0SP1:func.3">C_Algebra_of_BoundedFunctions</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">X</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E2:26"/><span class="lab"><font color="Green" title="E17">A2</font></span>: 
 <a href="cc0sp2.html#K3" title="CC0SP2:func.3">C_Algebra_of_ContinuousFunctions</a> <font color="Maroon" title="c1">X</font> is    <a href="cc0sp1.html#M1" title="CC0SP1:mode.1">ComplexSubAlgebra</a> of  <a href="cfuncdom.html#K8" title="CFUNCDOM:func.8">CAlgebra</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">X</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="cc0sp1.html#T2" title="CC0SP1:th.2">CC0SP1:2</a></span>;<br/>
<a NAME="E3:26"/>
 <a href="cc0sp1.html#K3" title="CC0SP1:func.3">C_Algebra_of_BoundedFunctions</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">X</font> is    <a href="cc0sp1.html#M1" title="CC0SP1:mode.1">ComplexSubAlgebra</a> of  <a href="cfuncdom.html#K8" title="CFUNCDOM:func.8">CAlgebra</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">X</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="cc0sp1.html#T4" title="CC0SP1:th.4">CC0SP1:4</a></span>;<br/>
<span class="kw">hence </span><a NAME="E4:26"/>
 <a href="cc0sp2.html#K3" title="CC0SP2:func.3">C_Algebra_of_ContinuousFunctions</a> <font color="Maroon" title="c1">X</font> is    <a href="cc0sp1.html#M1" title="CC0SP1:mode.1">ComplexSubAlgebra</a> of  <a href="cc0sp1.html#K3" title="CC0SP1:func.3">C_Algebra_of_BoundedFunctions</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">X</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:26"><span class="lab"><font color="Green" title="E16">A1</font></span></a>, <a class="txt" href="#E2:26"><span class="lab"><font color="Green" title="E17">A2</font></span></a>, <a class="ref" href="cc0sp2.html#T15" target="_self" title="CC0SP2:th.15">Th15</a></span>; <a class="txt" 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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