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<span class="kw">let </span><font color="Maroon" title="c1">A</font> be   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="measure5.html#V2" title="MEASURE5:attr.2">closed_interval</a>  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</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">u</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a> holds <br/> ( <font color="Olive" title="b1">u</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="c0sp2.html#K5" title="C0SP2:func.5">R_Normed_Algebra_of_ContinuousFunctions</a> <span class="p2">(<span class="default"><a href="dualsp05.html#K1" title="DUALSP05:func.1">ClstoCmp</a> <font color="Maroon" title="c1">A</font></span>)</span></span>)</span> iff (  <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <font color="Olive" title="b1">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">A</font> &amp; <font color="Olive" title="b1">u</font> is   <a href="fcont_1.html#V1" title="FCONT_1:attr.1">continuous</a>  <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>,<a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> ) )</span><br/><span class="kw">let </span><font color="Maroon" title="c2">u</font> be   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</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"> ( <font color="Maroon" title="c2">u</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="c0sp2.html#K5" title="C0SP2:func.5">R_Normed_Algebra_of_ContinuousFunctions</a> <span class="p2">(<span class="default"><a href="dualsp05.html#K1" title="DUALSP05:func.1">ClstoCmp</a> <font color="Maroon" title="c1">A</font></span>)</span></span>)</span> iff (  <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <font color="Maroon" title="c2">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">A</font> &amp; <font color="Maroon" title="c2">u</font> is   <a href="fcont_1.html#V1" title="FCONT_1:attr.1">continuous</a>  <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>,<a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> ) )</span><br/>



<span class="kw">consider </span><font color="Maroon" title="c3">a</font>, <font color="Maroon" title="c4">b</font> being   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a><span class="kw"> such that </span><br/><a NAME="E2:7"/><span class="lab"><font color="Green" title="E8">A1</font></span>: 
( <font color="Maroon" title="c3">a</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">b</font> &amp; <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">b</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">A</font> &amp;  <a href="dualsp05.html#K1" title="DUALSP05:func.1">ClstoCmp</a> <font color="Maroon" title="c1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="topmetr.html#K4" title="TOPMETR:func.4">Closed-Interval-TSpace</a> (<font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">b</font>) )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="dualsp05.html#D1" target="_self" onmouseover="rs('dualsp05/D1')" onmouseout="rh()">Def7ClstoCmp</a></span>;<br/>
<div><span class="kw">hereby </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"> (  <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <font color="Maroon" title="c2">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">A</font> &amp; <font color="Maroon" title="c2">u</font> is   <a href="fcont_1.html#V1" title="FCONT_1:attr.1">continuous</a>  <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>,<a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> implies <font color="Maroon" title="c2">u</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="c0sp2.html#K5" title="C0SP2:func.5">R_Normed_Algebra_of_ContinuousFunctions</a> <span class="p2">(<span class="default"><a href="dualsp05.html#K1" title="DUALSP05:func.1">ClstoCmp</a> <font color="Maroon" title="c1">A</font></span>)</span></span>)</span> )</span>
<div class="add"><span class="kw">assume </span><a NAME="E1:7_1"/>
<font color="Maroon" title="c2">u</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="c0sp2.html#K5" title="C0SP2:func.5">R_Normed_Algebra_of_ContinuousFunctions</a> <span class="p2">(<span class="default"><a href="dualsp05.html#K1" title="DUALSP05:func.1">ClstoCmp</a> <font color="Maroon" title="c1">A</font></span>)</span></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"> (  <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <font color="Maroon" title="c2">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">A</font> &amp; <font color="Maroon" title="c2">u</font> is   <a href="fcont_1.html#V1" title="FCONT_1:attr.1">continuous</a>  <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>,<a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> )</span><br/><a NAME="E2:7_1"/><span class="kw">then </span>
<font color="Maroon" title="c2">u</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="c0sp2.html#K2" title="C0SP2:func.2">ContinuousFunctions</a> <span class="p1">(<span class="default"><a href="dualsp05.html#K1" title="DUALSP05:func.1">ClstoCmp</a> <font color="Maroon" title="c1">A</font></span>)</span>
 ;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c5">w</font> being   <a href="pscomp_1.html#V1" title="PSCOMP_1:attr.1">continuous</a>  <a href="pscomp_1.html#NM1" title="PSCOMP_1:NM.1">RealMap</a> of <span class="p1">(<span class="default"><a href="dualsp05.html#K1" title="DUALSP05:func.1">ClstoCmp</a> <font color="Maroon" title="c1">A</font></span>)</span><span class="kw"> such that </span><br/><a NAME="E4:7_1"/><span class="lab"><font color="Green" title="E9">A2</font></span>: 
<font color="Maroon" title="c2">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">w</font>
 ;<br/><a NAME="E5:7_1"/><span class="lab"><font color="Green" title="E10">B3</font></span>: 
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c5">w</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="dualsp05.html#K1" title="DUALSP05:func.1">ClstoCmp</a> <font color="Maroon" title="c1">A</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D1" onmouseover="rs('funct_2/D1')" onmouseout="rh()">FUNCT_2:def 1</a></span>;<br/><a NAME="E6:7_1"/><span class="kw">then </span>
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c5">w</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">A</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="dualsp05.html#T5" target="_self" onmouseover="rs('dualsp05/T5')" onmouseout="rh()">Lm1</a></span>;<br/><a NAME="E7:7_1"/><span class="kw">then </span>
(  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c5">w</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  &amp;  <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c5">w</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  )
 ;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c6">v</font> = <font color="Maroon" title="c5">w</font> as   <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>,<a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> <span class="kw">by</span> <span class="lab"><a class="ref" href="relset_1.html#T4" onmouseover="rs('relset_1/T4')" onmouseout="rh()">RELSET_1:4</a></span>;<br/><a NAME="E9:7_1"/>
<font color="Maroon" title="c6">v</font> is   <a href="fcont_1.html#V1" title="FCONT_1:attr.1">continuous</a>  <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>,<a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:7"><span class="lab"><font color="Green" title="E8">A1</font></span></a>, <a class="ref" href="dualsp05.html#T4" target="_self" onmouseover="rs('dualsp05/T4')" onmouseout="rh()">Th80b</a></span>;<br/><span class="kw">hence </span><a NAME="E10:7_1"/>
(  <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <font color="Maroon" title="c2">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">A</font> &amp; <font color="Maroon" title="c2">u</font> is   <a href="fcont_1.html#V1" title="FCONT_1:attr.1">continuous</a>  <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>,<a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:7_1"><span class="lab"><font color="Green" title="E9">A2</font></span></a>, <a class="txt" href="#E5:7_1"><span class="lab"><font color="Green" title="E10">B3</font></span></a>, <a class="ref" href="dualsp05.html#T5" target="_self" onmouseover="rs('dualsp05/T5')" onmouseout="rh()">Lm1</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/></div>
<span class="kw">end;</span></div>

<span class="kw">assume </span><a NAME="E4:7"/><span class="lab"><font color="Green" title="E9">A4</font></span>: 
(  <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <font color="Maroon" title="c2">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">A</font> &amp; <font color="Maroon" title="c2">u</font> is   <a href="fcont_1.html#V1" title="FCONT_1:attr.1">continuous</a>  <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>,<a href="numbers.html#K1" title="NUMBERS:func.1">REAL</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"> <font color="Maroon" title="c2">u</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="c0sp2.html#K5" title="C0SP2:func.5">R_Normed_Algebra_of_ContinuousFunctions</a> <span class="p2">(<span class="default"><a href="dualsp05.html#K1" title="DUALSP05:func.1">ClstoCmp</a> <font color="Maroon" title="c1">A</font></span>)</span></span>)</span></span><br/>

<a NAME="E5:7"/><span class="kw">then </span><span class="lab"><font color="Green" title="E10">A5</font></span>: 
 <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <font color="Maroon" title="c2">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="dualsp05.html#K1" title="DUALSP05:func.1">ClstoCmp</a> <font color="Maroon" title="c1">A</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="dualsp05.html#T5" target="_self" onmouseover="rs('dualsp05/T5')" onmouseout="rh()">Lm1</a></span>;<br/>
<a NAME="E6:7"/>
 <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <font color="Maroon" title="c2">u</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:7"><span class="lab"><font color="Green" title="E9">A4</font></span></a>, <a class="ref" href="relat_1.html#D19" onmouseover="rs('relat_1/D19')" onmouseout="rh()">RELAT_1:def 19</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c5">v</font> = <font color="Maroon" title="c2">u</font> as   <a href="pscomp_1.html#NM1" title="PSCOMP_1:NM.1">RealMap</a> of <span class="p1">(<span class="default"><a href="dualsp05.html#K1" title="DUALSP05:func.1">ClstoCmp</a> <font color="Maroon" title="c1">A</font></span>)</span> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:7"><span class="lab"><font color="Green" title="E10">A5</font></span></a>, <a class="ref" href="funct_2.html#T2" onmouseover="rs('funct_2/T2')" onmouseout="rh()">FUNCT_2:2</a></span>;<br/>
<a NAME="E8:7"/>
<font color="Maroon" title="c5">v</font> is  <a href="pscomp_1.html#V1" title="PSCOMP_1:attr.1">continuous</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:7"><span class="lab"><font color="Green" title="E8">A1</font></span></a>, <a class="txt" href="#E4:7"><span class="lab"><font color="Green" title="E9">A4</font></span></a>, <a class="ref" href="dualsp05.html#T4" target="_self" onmouseover="rs('dualsp05/T4')" onmouseout="rh()">Th80b</a></span>;<br/>
<a NAME="E9:7"/><span class="kw">then </span>
<font color="Maroon" title="c5">v</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="c0sp2.html#K5" title="C0SP2:func.5">R_Normed_Algebra_of_ContinuousFunctions</a> <span class="p2">(<span class="default"><a href="dualsp05.html#K1" title="DUALSP05:func.1">ClstoCmp</a> <font color="Maroon" title="c1">A</font></span>)</span></span>)</span>
 
;<br/>
<span class="kw">hence </span><a NAME="E10:7"/>
<font color="Maroon" title="c2">u</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="c0sp2.html#K5" title="C0SP2:func.5">R_Normed_Algebra_of_ContinuousFunctions</a> <span class="p2">(<span class="default"><a href="dualsp05.html#K1" title="DUALSP05:func.1">ClstoCmp</a> <font color="Maroon" title="c1">A</font></span>)</span></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"> verum</span><br/>


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