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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">rho</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c1">A</font>,<a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a><br/>  for <font color="Olive" title="b2">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="dualsp01.html#K17" title="DUALSP01:func.17">DualSp</a> <span class="p2">(<span class="default"><a href="c0sp2.html#K5" title="C0SP2:func.5">R_Normed_Algebra_of_ContinuousFunctions</a> <span class="p3">(<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>)</span>  st <font color="Olive" title="b1">rho</font> is  <a href="integr22.html#V1" title="INTEGR22:attr.1">bounded_variation</a>  &amp; (  for <font color="Olive" title="b3">u</font> being   <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>  st  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Olive" title="b3">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">A</font> holds <br/><font color="Olive" title="b2">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b3">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="integr22.html#K5" title="INTEGR22:func.5">integral</a> (<font color="Olive" title="b3">u</font>,<font color="Olive" title="b1">rho</font>) ) holds <br/><span class="p1"><a href="normsp_0.html#K2" title="NORMSP_0:func.2">||.</a><span class="default"><font color="Olive" title="b2">f</font></span><a href="normsp_0.html#K2" title="NORMSP_0:func.2">.||</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="integr22.html#K2" title="INTEGR22:func.2">total_vd</a> <font color="Olive" title="b1">rho</font></span><br/><span class="kw">let </span><font color="Maroon" title="c2">rho</font> be   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c1">A</font>,<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">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="dualsp01.html#K17" title="DUALSP01:func.17">DualSp</a> <span class="p2">(<span class="default"><a href="c0sp2.html#K5" title="C0SP2:func.5">R_Normed_Algebra_of_ContinuousFunctions</a> <span class="p3">(<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>)</span>  st <font color="Maroon" title="c2">rho</font> is  <a href="integr22.html#V1" title="INTEGR22:attr.1">bounded_variation</a>  &amp; (  for <font color="Olive" title="b2">u</font> being   <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>  st  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Olive" title="b2">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">A</font> holds <br/><font color="Olive" title="b1">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="integr22.html#K5" title="INTEGR22:func.5">integral</a> (<font color="Olive" title="b2">u</font>,<font color="Maroon" title="c2">rho</font>) ) holds <br/><span class="p1"><a href="normsp_0.html#K2" title="NORMSP_0:func.2">||.</a><span class="default"><font color="Olive" title="b1">f</font></span><a href="normsp_0.html#K2" title="NORMSP_0:func.2">.||</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="integr22.html#K2" title="INTEGR22:func.2">total_vd</a> <font color="Maroon" title="c2">rho</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="dualsp01.html#K17" title="DUALSP01:func.17">DualSp</a> <span class="p2">(<span class="default"><a href="c0sp2.html#K5" title="C0SP2:func.5">R_Normed_Algebra_of_ContinuousFunctions</a> <span class="p3">(<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>)</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"> ( <font color="Maroon" title="c2">rho</font> is  <a href="integr22.html#V1" title="INTEGR22:attr.1">bounded_variation</a>  &amp; (  for <font color="Olive" title="b1">u</font> being   <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>  st  <a href="relset_1.html#K1" title="RELSET_1:func.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> holds <br/><font color="Maroon" title="c3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="integr22.html#K5" title="INTEGR22:func.5">integral</a> (<font color="Olive" title="b1">u</font>,<font color="Maroon" title="c2">rho</font>) ) implies <span class="p1"><a href="normsp_0.html#K2" title="NORMSP_0:func.2">||.</a><span class="default"><font color="Maroon" title="c3">f</font></span><a href="normsp_0.html#K2" title="NORMSP_0:func.2">.||</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="integr22.html#K2" title="INTEGR22:func.2">total_vd</a> <font color="Maroon" title="c2">rho</font> )</span><br/>





<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:21"/><span class="lab"><font color="Green" title="E22">A1</font></span>: 
<font color="Maroon" title="c2">rho</font> is  <a href="integr22.html#V1" title="INTEGR22:attr.1">bounded_variation</a> 
 <span class="kw">and </span><br/><a NAME="E2:21"/><span class="lab"><font color="Green" title="E23">A2</font></span>: 
 for <font color="Olive" title="b1">u</font> being   <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>  st  <a href="relset_1.html#K1" title="RELSET_1:func.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> holds <br/><font color="Maroon" title="c3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="integr22.html#K5" title="INTEGR22:func.5">integral</a> (<font color="Olive" title="b1">u</font>,<font color="Maroon" title="c2">rho</font>)
 ; <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"><a href="normsp_0.html#K2" title="NORMSP_0:func.2">||.</a><span class="default"><font color="Maroon" title="c3">f</font></span><a href="normsp_0.html#K2" title="NORMSP_0:func.2">.||</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="integr22.html#K2" title="INTEGR22:func.2">total_vd</a> <font color="Maroon" title="c2">rho</font></span><br/>

<span class="kw">set </span><font color="Maroon" title="c4">X</font> =  <a href="c0sp2.html#K5" title="C0SP2:func.5">R_Normed_Algebra_of_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/>
<a NAME="E3:21"/><span class="lab"><font color="Green" title="E24">A3</font></span>: 
 for <font color="Olive" title="b1">u</font> being   <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>  st <font color="Olive" title="b1">u</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> holds <br/><font color="Maroon" title="c3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="integr22.html#K5" title="INTEGR22:func.5">integral</a> (<font color="Olive" title="b1">u</font>,<font color="Maroon" title="c2">rho</font>)
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="21_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c5">u</font> be   <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="c5">u</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> implies <font color="Maroon" title="c3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="integr22.html#K5" title="INTEGR22:func.5">integral</a> (<font color="Maroon" title="c5">u</font>,<font color="Maroon" title="c2">rho</font>) )</span><br/>

<span class="kw">assume </span><a NAME="E1:21_1"/>
<font color="Maroon" title="c5">u</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>
 ; <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="c3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="integr22.html#K5" title="INTEGR22:func.5">integral</a> (<font color="Maroon" title="c5">u</font>,<font color="Maroon" title="c2">rho</font>)</span><br/>

<a NAME="E2:21_1"/><span class="kw">then </span>
(  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c5">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">A</font> &amp; <font color="Maroon" title="c5">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="ref" href="dualsp05.html#T6" target="_self" onmouseover="rs('dualsp05/T6')" onmouseout="rh()">Th80</a></span>;<br/>
<span class="kw">hence </span><a NAME="E3:21_1"/>
<font color="Maroon" title="c3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="integr22.html#K5" title="INTEGR22:func.5">integral</a> (<font color="Maroon" title="c5">u</font>,<font color="Maroon" title="c2">rho</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:21"><span class="lab"><font color="Green" title="E23">A2</font></span></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>
<a NAME="E4:21"/><span class="lab"><font color="Green" title="E25">A4</font></span>: 
 for <font color="Olive" title="b1">u</font> being   <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><br/>  for <font color="Olive" title="b2">v</font> being   <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>  st  <a href="relset_1.html#K1" title="RELSET_1:func.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> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">v</font> holds <br/><span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><a href="integr22.html#K5" title="INTEGR22:func.5">integral</a> (<font color="Olive" title="b1">u</font>,<font color="Maroon" title="c2">rho</font>)</span>)</span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1"><a href="normsp_0.html#K2" title="NORMSP_0:func.2">||.</a><span class="default"><font color="Olive" title="b2">v</font></span><a href="normsp_0.html#K2" title="NORMSP_0:func.2">.||</a></span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><a href="integr22.html#K2" title="INTEGR22:func.2">total_vd</a> <font color="Maroon" title="c2">rho</font></span>)</span>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="21_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c5">u</font> be   <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">  for <font color="Olive" title="b1">v</font> being   <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>  st  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c5">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">A</font> &amp; <font color="Maroon" title="c5">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">v</font> holds <br/><span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><a href="integr22.html#K5" title="INTEGR22:func.5">integral</a> (<font color="Maroon" title="c5">u</font>,<font color="Maroon" title="c2">rho</font>)</span>)</span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1"><a href="normsp_0.html#K2" title="NORMSP_0:func.2">||.</a><span class="default"><font color="Olive" title="b1">v</font></span><a href="normsp_0.html#K2" title="NORMSP_0:func.2">.||</a></span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><a href="integr22.html#K2" title="INTEGR22:func.2">total_vd</a> <font color="Maroon" title="c2">rho</font></span>)</span></span><br/><span class="kw">let </span><font color="Maroon" title="c6">v</font> be   <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="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c5">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">A</font> &amp; <font color="Maroon" title="c5">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">v</font> implies <span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><a href="integr22.html#K5" title="INTEGR22:func.5">integral</a> (<font color="Maroon" title="c5">u</font>,<font color="Maroon" title="c2">rho</font>)</span>)</span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1"><a href="normsp_0.html#K2" title="NORMSP_0:func.2">||.</a><span class="default"><font color="Maroon" title="c6">v</font></span><a href="normsp_0.html#K2" title="NORMSP_0:func.2">.||</a></span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><a href="integr22.html#K2" title="INTEGR22:func.2">total_vd</a> <font color="Maroon" title="c2">rho</font></span>)</span> )</span><br/>



<span class="kw">assume </span><a NAME="E1:21_2"/><span class="lab"><font color="Green" title="E26">A5</font></span>: 
(  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c5">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">A</font> &amp; <font color="Maroon" title="c5">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">v</font> )
 ; <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"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><a href="integr22.html#K5" title="INTEGR22:func.5">integral</a> (<font color="Maroon" title="c5">u</font>,<font color="Maroon" title="c2">rho</font>)</span>)</span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1"><a href="normsp_0.html#K2" title="NORMSP_0:func.2">||.</a><span class="default"><font color="Maroon" title="c6">v</font></span><a href="normsp_0.html#K2" title="NORMSP_0:func.2">.||</a></span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><a href="integr22.html#K2" title="INTEGR22:func.2">total_vd</a> <font color="Maroon" title="c2">rho</font></span>)</span></span><br/>

<a NAME="E2:21_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E27">B6</font></span>: 
<font color="Maroon" title="c5">u</font> <a href="integr22.html#R1" title="INTEGR22:pred.1">is_RiemannStieltjes_integrable_with</a> <font color="Maroon" title="c2">rho</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:21"><span class="lab"><font color="Green" title="E22">A1</font></span></a>, <a class="ref" href="integr23.html#T21" onmouseover="rs('integr23/T21')" onmouseout="rh()">INTEGR23:21</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c7">T</font> being   <a href="integra2.html#NM1" title="INTEGRA2:NM.1">DivSequence</a> of <font color="Maroon" title="c1">A</font><span class="kw"> such that </span><br/><a NAME="E4:21_2"/><span class="lab"><font color="Green" title="E28">A7</font></span>: 
(  <a href="integra3.html#K2" title="INTEGRA3:func.2">delta</a> <font color="Maroon" title="c7">T</font> is  <a href="comseq_2.html#V2" title="COMSEQ_2:attr.2">convergent</a>  &amp;  <a href="seq_2.html#K1" title="SEQ_2:func.1">lim</a> <span class="p1">(<span class="default"><a href="integra3.html#K2" title="INTEGRA3:func.2">delta</a> <font color="Maroon" title="c7">T</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="integra4.html#T11" onmouseover="rs('integra4/T11')" onmouseout="rh()">INTEGRA4:11</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c8">S</font> =  the    <a href="integr22.html#M3" title="INTEGR22:mode.3">middle_volume_Sequence</a> of <font color="Maroon" title="c2">rho</font>,<font color="Maroon" title="c5">u</font>,<font color="Maroon" title="c7">T</font>;<br/>
<a NAME="E5:21_2"/><span class="lab"><font color="Green" title="E29">A9</font></span>: 
<span class="p1"><a href="valued_1.html#K55" title="VALUED_1:func.55">|.</a><span class="default"><span class="p2">(<span class="default"><a href="integr22.html#K4" title="INTEGR22:func.4">middle_sum</a>  the    <a href="integr22.html#M3" title="INTEGR22:mode.3">middle_volume_Sequence</a> of <font color="Maroon" title="c2">rho</font>,<font color="Maroon" title="c5">u</font>,<font color="Maroon" title="c7">T</font></span>)</span></span><a href="valued_1.html#K55" title="VALUED_1:func.55">.|</a></span> is  <a href="comseq_2.html#V2" title="COMSEQ_2:attr.2">convergent</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="seq_4.html#T13" onmouseover="rs('seq_4/T13')" onmouseout="rh()">SEQ_4:13</a>, <a class="txt" href="#E2:21_2"><span class="lab"><font color="Green" title="E27">B6</font></span></a>, <a class="txt" href="#E4:21_2"><span class="lab"><font color="Green" title="E28">A7</font></span></a></span>;<br/>
<a NAME="E6:21_2"/><span class="lab"><font color="Green" title="E30">A10</font></span>: 
 for <font color="Olive" title="b1">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> holds  <span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="integr22.html#K4" title="INTEGR22:func.4">middle_sum</a>  the    <a href="integr22.html#M3" title="INTEGR22:mode.3">middle_volume_Sequence</a> of <font color="Maroon" title="c2">rho</font>,<font color="Maroon" title="c5">u</font>,<font color="Maroon" title="c7">T</font></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Olive" title="b1">n</font></span>)</span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1"><a href="normsp_0.html#K2" title="NORMSP_0:func.2">||.</a><span class="default"><font color="Maroon" title="c6">v</font></span><a href="normsp_0.html#K2" title="NORMSP_0:func.2">.||</a></span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><a href="integr22.html#K2" title="INTEGR22:func.2">total_vd</a> <font color="Maroon" title="c2">rho</font></span>)</span>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="21_2_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c9">n</font> be   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</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"> <span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="integr22.html#K4" title="INTEGR22:func.4">middle_sum</a>  the    <a href="integr22.html#M3" title="INTEGR22:mode.3">middle_volume_Sequence</a> of <font color="Maroon" title="c2">rho</font>,<font color="Maroon" title="c5">u</font>,<font color="Maroon" title="c7">T</font></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c9">n</font></span>)</span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1"><a href="normsp_0.html#K2" title="NORMSP_0:func.2">||.</a><span class="default"><font color="Maroon" title="c6">v</font></span><a href="normsp_0.html#K2" title="NORMSP_0:func.2">.||</a></span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><a href="integr22.html#K2" title="INTEGR22:func.2">total_vd</a> <font color="Maroon" title="c2">rho</font></span>)</span></span><br/>

<span class="kw">set </span><font color="Maroon" title="c10">F</font> =  the    <a href="integr22.html#M1" title="INTEGR22:mode.1">var_volume</a> of <font color="Maroon" title="c2">rho</font>,<font color="Maroon" title="c7">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c9">n</font>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c11">v0</font> = <span class="p1"><a href="normsp_0.html#K2" title="NORMSP_0:func.2">||.</a><span class="default"><font color="Maroon" title="c6">v</font></span><a href="normsp_0.html#K2" title="NORMSP_0:func.2">.||</a></span> as   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a> ;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c12">vF</font> = <font color="Maroon" title="c11">v0</font> <a href="rvsum_1.html#K10" title="RVSUM_1:func.10">*</a>  the    <a href="integr22.html#M1" title="INTEGR22:mode.1">var_volume</a> of <font color="Maroon" title="c2">rho</font>,<font color="Maroon" title="c7">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c9">n</font> as    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  ;<br/>
<a NAME="E3:21_2_1"/>
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a>  the    <a href="integr22.html#M1" title="INTEGR22:mode.1">var_volume</a> of <font color="Maroon" title="c2">rho</font>,<font color="Maroon" title="c7">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c9">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a>  the    <a href="integr22.html#M1" title="INTEGR22:mode.1">var_volume</a> of <font color="Maroon" title="c2">rho</font>,<font color="Maroon" title="c7">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c9">n</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D3" onmouseover="rs('finseq_1/D3')" onmouseout="rh()">FINSEQ_1:def 3</a></span>;<br/>
<a NAME="E4:21_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E31">B14</font></span>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c12">vF</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a>  the    <a href="integr22.html#M1" title="INTEGR22:mode.1">var_volume</a> of <font color="Maroon" title="c2">rho</font>,<font color="Maroon" title="c7">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c9">n</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="valued_1.html#D5" onmouseover="rs('valued_1/D5')" onmouseout="rh()">VALUED_1:def 5</a></span>;<br/>
<span class="lab"><font color="Green" title="E32">A15</font></span>:  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"> the    <a href="integr22.html#M3" title="INTEGR22:mode.3">middle_volume_Sequence</a> of <font color="Maroon" title="c2">rho</font>,<font color="Maroon" title="c5">u</font>,<font color="Maroon" title="c7">T</font> <a href="integr22.html#K3" title="INTEGR22:func.3">.</a> <font color="Maroon" title="c9">n</font></span>)</span> = 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c9">n</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:21"><span class="lab"><font color="Green" title="E22">A1</font></span></a>, <a class="txt" href="#E1:21_2"><span class="lab"><font color="Green" title="E26">A5</font></span></a>, <a class="ref" href="integr22.html#D5" onmouseover="rs('integr22/D5')" onmouseout="rh()">INTEGR22:def 5</a></span>
<br/>.= 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a>  the    <a href="integr22.html#M1" title="INTEGR22:mode.1">var_volume</a> of <font color="Maroon" title="c2">rho</font>,<font color="Maroon" title="c7">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c9">n</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="integr22.html#D2" onmouseover="rs('integr22/D2')" onmouseout="rh()">INTEGR22:def 2</a></span>
<br/>.= 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c12">vF</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:21_2_1"><span class="lab"><font color="Green" title="E31">B14</font></span></a>, <a class="ref" href="finseq_1.html#D3" onmouseover="rs('finseq_1/D3')" onmouseout="rh()">FINSEQ_1:def 3</a></span>
;<br/>
<a NAME="E6:21_2_1"/>
 for <font color="Olive" title="b1">j</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Olive" title="b1">j</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"> the    <a href="integr22.html#M3" title="INTEGR22:mode.3">middle_volume_Sequence</a> of <font color="Maroon" title="c2">rho</font>,<font color="Maroon" title="c5">u</font>,<font color="Maroon" title="c7">T</font> <a href="integr22.html#K3" title="INTEGR22:func.3">.</a> <font color="Maroon" title="c9">n</font></span>)</span> holds <br/><span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"> the    <a href="integr22.html#M3" title="INTEGR22:mode.3">middle_volume_Sequence</a> of <font color="Maroon" title="c2">rho</font>,<font color="Maroon" title="c5">u</font>,<font color="Maroon" title="c7">T</font> <a href="integr22.html#K3" title="INTEGR22:func.3">.</a> <font color="Maroon" title="c9">n</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">j</font></span>)</span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c12">vF</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">j</font>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="21_2_1_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c13">j</font> be   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</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="c13">j</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"> the    <a href="integr22.html#M3" title="INTEGR22:mode.3">middle_volume_Sequence</a> of <font color="Maroon" title="c2">rho</font>,<font color="Maroon" title="c5">u</font>,<font color="Maroon" title="c7">T</font> <a href="integr22.html#K3" title="INTEGR22:func.3">.</a> <font color="Maroon" title="c9">n</font></span>)</span> implies <span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"> the    <a href="integr22.html#M3" title="INTEGR22:mode.3">middle_volume_Sequence</a> of <font color="Maroon" title="c2">rho</font>,<font color="Maroon" title="c5">u</font>,<font color="Maroon" title="c7">T</font> <a href="integr22.html#K3" title="INTEGR22:func.3">.</a> <font color="Maroon" title="c9">n</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">j</font></span>)</span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c12">vF</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">j</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:21_2_1_2"/>
<font color="Maroon" title="c13">j</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"> the    <a href="integr22.html#M3" title="INTEGR22:mode.3">middle_volume_Sequence</a> of <font color="Maroon" title="c2">rho</font>,<font color="Maroon" title="c5">u</font>,<font color="Maroon" title="c7">T</font> <a href="integr22.html#K3" title="INTEGR22:func.3">.</a> <font color="Maroon" title="c9">n</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"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"> the    <a href="integr22.html#M3" title="INTEGR22:mode.3">middle_volume_Sequence</a> of <font color="Maroon" title="c2">rho</font>,<font color="Maroon" title="c5">u</font>,<font color="Maroon" title="c7">T</font> <a href="integr22.html#K3" title="INTEGR22:func.3">.</a> <font color="Maroon" title="c9">n</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">j</font></span>)</span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c12">vF</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">j</font></span><br/>

<a NAME="E2:21_2_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E33">B17</font></span>: 
<font color="Maroon" title="c13">j</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"> the    <a href="integr22.html#M3" title="INTEGR22:mode.3">middle_volume_Sequence</a> of <font color="Maroon" title="c2">rho</font>,<font color="Maroon" title="c5">u</font>,<font color="Maroon" title="c7">T</font> <a href="integr22.html#K3" title="INTEGR22:func.3">.</a> <font color="Maroon" title="c9">n</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D3" onmouseover="rs('finseq_1/D3')" onmouseout="rh()">FINSEQ_1:def 3</a></span>;<br/>
<a NAME="E3:21_2_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c13">j</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c9">n</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:21"><span class="lab"><font color="Green" title="E22">A1</font></span></a>, <a class="txt" href="#E1:21_2"><span class="lab"><font color="Green" title="E26">A5</font></span></a>, <a class="ref" href="integr22.html#D5" onmouseover="rs('integr22/D5')" onmouseout="rh()">INTEGR22:def 5</a></span>;<br/>
<a NAME="E4:21_2_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E34">A16</font></span>: 
<font color="Maroon" title="c13">j</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c9">n</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D3" onmouseover="rs('finseq_1/D3')" onmouseout="rh()">FINSEQ_1:def 3</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c14">r</font> being   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a><span class="kw"> such that </span><br/><a NAME="E6:21_2_1_2"/><span class="lab"><font color="Green" title="E35">A18</font></span>: 
<font color="Maroon" title="c14">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">u</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<span class="p3">(<span class="default"><font color="Maroon" title="c7">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c9">n</font></span>)</span>,<font color="Maroon" title="c13">j</font>)</span>)</span></span>)</span>
 <span class="kw">and </span><br/><a NAME="E7:21_2_1_2"/><span class="lab"><font color="Green" title="E36">A19</font></span>: 
<span class="p1">(<span class="default"> the    <a href="integr22.html#M3" title="INTEGR22:mode.3">middle_volume_Sequence</a> of <font color="Maroon" title="c2">rho</font>,<font color="Maroon" title="c5">u</font>,<font color="Maroon" title="c7">T</font> <a href="integr22.html#K3" title="INTEGR22:func.3">.</a> <font color="Maroon" title="c9">n</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">r</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><a href="integr22.html#K1" title="INTEGR22:func.1">vol</a> (<span class="p2">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<span class="p3">(<span class="default"><font color="Maroon" title="c7">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c9">n</font></span>)</span>,<font color="Maroon" title="c13">j</font>)</span>)</span>,<font color="Maroon" title="c2">rho</font>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:21"><span class="lab"><font color="Green" title="E22">A1</font></span></a>, <a class="txt" href="#E1:21_2"><span class="lab"><font color="Green" title="E26">A5</font></span></a>, <a class="ref" href="integr22.html#D5" onmouseover="rs('integr22/D5')" onmouseout="rh()">INTEGR22:def 5</a>, <a class="txt" href="#E4:21_2_1_2"><span class="lab"><font color="Green" title="E34">A16</font></span></a></span>;<br/>
<span class="lab"><font color="Green" title="E37">A20</font></span>: <span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"> the    <a href="integr22.html#M3" title="INTEGR22:mode.3">middle_volume_Sequence</a> of <font color="Maroon" title="c2">rho</font>,<font color="Maroon" title="c5">u</font>,<font color="Maroon" title="c7">T</font> <a href="integr22.html#K3" title="INTEGR22:func.3">.</a> <font color="Maroon" title="c9">n</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">j</font></span>)</span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span> = 
<span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><font color="Maroon" title="c14">r</font></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><a href="integr22.html#K1" title="INTEGR22:func.1">vol</a> (<span class="p3">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<span class="p4">(<span class="default"><font color="Maroon" title="c7">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c9">n</font></span>)</span>,<font color="Maroon" title="c13">j</font>)</span>)</span>,<font color="Maroon" title="c2">rho</font>)</span>)</span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="complex1.html#T65" onmouseover="rs('complex1/T65')" onmouseout="rh()">COMPLEX1:65</a>, <a class="txt" href="#E7:21_2_1_2"><span class="lab"><font color="Green" title="E36">A19</font></span></a></span>
<br/>.= 
<span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><font color="Maroon" title="c14">r</font></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"> the    <a href="integr22.html#M1" title="INTEGR22:mode.1">var_volume</a> of <font color="Maroon" title="c2">rho</font>,<font color="Maroon" title="c7">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c9">n</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">j</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:21_2_1_2"><span class="lab"><font color="Green" title="E34">A16</font></span></a>, <a class="ref" href="integr22.html#D2" onmouseover="rs('integr22/D2')" onmouseout="rh()">INTEGR22:def 2</a></span>
;<br/>
<span class="kw">consider </span><font color="Maroon" title="c15">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> <span class="kw"> such that </span><br/><a NAME="E10:21_2_1_2"/><span class="lab"><font color="Green" title="E38">A21</font></span>: 
( <font color="Maroon" title="c15">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">u</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<span class="p3">(<span class="default"><font color="Maroon" title="c7">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c9">n</font></span>)</span>,<font color="Maroon" title="c13">j</font>)</span>)</span></span>)</span> &amp; <font color="Maroon" title="c14">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">u</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<span class="p3">(<span class="default"><font color="Maroon" title="c7">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c9">n</font></span>)</span>,<font color="Maroon" title="c13">j</font>)</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">x</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:21_2_1_2"><span class="lab"><font color="Green" title="E35">A18</font></span></a>, <a class="ref" href="funct_1.html#D3" onmouseover="rs('funct_1/D3')" onmouseout="rh()">FUNCT_1:def 3</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c16">AV</font> =  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>;<br/>
<a NAME="E11:21_2_1_2"/>
<font color="Maroon" title="c5">u</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="c0sp1.html#K6" title="C0SP1:func.6">BoundedFunctions</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="txt" href="#E1:21_2"><span class="lab"><font color="Green" title="E26">A5</font></span></a>, <a class="ref" href="dualsp05.html#T7" target="_self" onmouseover="rs('dualsp05/T7')" onmouseout="rh()">Lm2</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c17">u1</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of  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>,<a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a><span class="kw"> such that </span><br/><a NAME="E13:21_2_1_2"/><span class="lab"><font color="Green" title="E39">A23</font></span>: 
( <font color="Maroon" title="c5">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">u1</font> &amp; <font color="Maroon" title="c17">u1</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</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> is  <a href="comseq_2.html#V1" title="COMSEQ_2:attr.1">bounded</a>  )
 ;<br/>
<a NAME="E14:21_2_1_2"/>
<font color="Maroon" title="c15">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:21_2"><span class="lab"><font color="Green" title="E26">A5</font></span></a>, <a class="txt" href="#E10:21_2_1_2"><span class="lab"><font color="Green" title="E38">A21</font></span></a>, <a class="ref" href="relat_1.html#T57" onmouseover="rs('relat_1/T57')" onmouseout="rh()">RELAT_1:57</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c18">x1</font> = <font color="Maroon" title="c15">x</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  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/>
<span class="kw">reconsider </span><font color="Maroon" title="c19">v1</font> = <font color="Maroon" title="c6">v</font> as   <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>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <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 class="kw">by</span> <span class="lab"><a class="ref" href="dualsp05.html#T7" target="_self" onmouseover="rs('dualsp05/T7')" onmouseout="rh()">Lm2</a></span>;<br/>
<a NAME="E17:21_2_1_2"/>
<span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c17">u1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c18">x1</font></span>)</span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1"><a href="normsp_0.html#K2" title="NORMSP_0:func.2">||.</a><span class="default"><font color="Maroon" title="c19">v1</font></span><a href="normsp_0.html#K2" title="NORMSP_0:func.2">.||</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:21_2"><span class="lab"><font color="Green" title="E26">A5</font></span></a>, <a class="txt" href="#E13:21_2_1_2"><span class="lab"><font color="Green" title="E39">A23</font></span></a>, <a class="ref" href="c0sp1.html#T26" onmouseover="rs('c0sp1/T26')" onmouseout="rh()">C0SP1:26</a></span>;<br/>
<a NAME="E18:21_2_1_2"/><span class="kw">then </span>
<span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">u</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">x</font></span>)</span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1"><a href="normsp_0.html#K2" title="NORMSP_0:func.2">||.</a><span class="default"><font color="Maroon" title="c6">v</font></span><a href="normsp_0.html#K2" title="NORMSP_0:func.2">.||</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:21_2_1_2"><span class="lab"><font color="Green" title="E39">A23</font></span></a>, <a class="ref" href="funct_1.html#T49" onmouseover="rs('funct_1/T49')" onmouseout="rh()">FUNCT_1:49</a></span>;<br/>
<a NAME="E19:21_2_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E40">A24</font></span>: 
<span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><font color="Maroon" title="c14">r</font></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1"><a href="normsp_0.html#K2" title="NORMSP_0:func.2">||.</a><span class="default"><font color="Maroon" title="c6">v</font></span><a href="normsp_0.html#K2" title="NORMSP_0:func.2">.||</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:21_2_1_2"><span class="lab"><font color="Green" title="E38">A21</font></span></a>, <a class="ref" href="funct_1.html#T47" onmouseover="rs('funct_1/T47')" onmouseout="rh()">FUNCT_1:47</a></span>;<br/>
<a NAME="E20:21_2_1_2"/>
<font color="Maroon" title="c13">j</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c9">n</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:21_2_1_2"><span class="lab"><font color="Green" title="E33">B17</font></span></a>, <a class="txt" href="#E1:21"><span class="lab"><font color="Green" title="E22">A1</font></span></a>, <a class="txt" href="#E1:21_2"><span class="lab"><font color="Green" title="E26">A5</font></span></a>, <a class="ref" href="integr22.html#D5" onmouseover="rs('integr22/D5')" onmouseout="rh()">INTEGR22:def 5</a></span>;<br/>
<a NAME="E21:21_2_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c13">j</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a>  the    <a href="integr22.html#M1" title="INTEGR22:mode.1">var_volume</a> of <font color="Maroon" title="c2">rho</font>,<font color="Maroon" title="c7">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c9">n</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="integr22.html#D2" onmouseover="rs('integr22/D2')" onmouseout="rh()">INTEGR22:def 2</a></span>;<br/>
<a NAME="E22:21_2_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c13">j</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a>  the    <a href="integr22.html#M1" title="INTEGR22:mode.1">var_volume</a> of <font color="Maroon" title="c2">rho</font>,<font color="Maroon" title="c7">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c9">n</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D3" onmouseover="rs('finseq_1/D3')" onmouseout="rh()">FINSEQ_1:def 3</a></span>;<br/>
<a NAME="E23:21_2_1_2"/><span class="kw">then </span>
 <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  the    <a href="integr22.html#M1" title="INTEGR22:mode.1">var_volume</a> of <font color="Maroon" title="c2">rho</font>,<font color="Maroon" title="c7">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c9">n</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">j</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="integr22.html#T3" onmouseover="rs('integr22/T3')" onmouseout="rh()">INTEGR22:3</a></span>;<br/>
<a NAME="E24:21_2_1_2"/><span class="kw">then </span>
<span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><font color="Maroon" title="c14">r</font></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"> the    <a href="integr22.html#M1" title="INTEGR22:mode.1">var_volume</a> of <font color="Maroon" title="c2">rho</font>,<font color="Maroon" title="c7">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c9">n</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">j</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c11">v0</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"> the    <a href="integr22.html#M1" title="INTEGR22:mode.1">var_volume</a> of <font color="Maroon" title="c2">rho</font>,<font color="Maroon" title="c7">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c9">n</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">j</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:21_2_1_2"><span class="lab"><font color="Green" title="E40">A24</font></span></a>, <a class="ref" href="xreal_1.html#T64" onmouseover="rs('xreal_1/T64')" onmouseout="rh()">XREAL_1:64</a></span>;<br/>
<span class="kw">hence </span><a NAME="E25:21_2_1_2"/>
<span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"> the    <a href="integr22.html#M3" title="INTEGR22:mode.3">middle_volume_Sequence</a> of <font color="Maroon" title="c2">rho</font>,<font color="Maroon" title="c5">u</font>,<font color="Maroon" title="c7">T</font> <a href="integr22.html#K3" title="INTEGR22:func.3">.</a> <font color="Maroon" title="c9">n</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">j</font></span>)</span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c12">vF</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">j</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:21_2_1_2"><span class="lab"><font color="Green" title="E37">A20</font></span></a>, <a class="ref" href="valued_1.html#T6" onmouseover="rs('valued_1/T6')" onmouseout="rh()">VALUED_1:6</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>
<a NAME="E7:21_2_1"/><span class="kw">then </span>
<span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p3">(<span class="default"> the    <a href="integr22.html#M3" title="INTEGR22:mode.3">middle_volume_Sequence</a> of <font color="Maroon" title="c2">rho</font>,<font color="Maroon" title="c5">u</font>,<font color="Maroon" title="c7">T</font> <a href="integr22.html#K3" title="INTEGR22:func.3">.</a> <font color="Maroon" title="c9">n</font></span>)</span></span>)</span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <font color="Maroon" title="c12">vF</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:21_2_1"><span class="lab"><font color="Green" title="E32">A15</font></span></a>, <a class="ref" href="integr23.html#T3" onmouseover="rs('integr23/T3')" onmouseout="rh()">INTEGR23:3</a></span>;<br/>
<a NAME="E8:21_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E33">A25</font></span>: 
<span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p3">(<span class="default"> the    <a href="integr22.html#M3" title="INTEGR22:mode.3">middle_volume_Sequence</a> of <font color="Maroon" title="c2">rho</font>,<font color="Maroon" title="c5">u</font>,<font color="Maroon" title="c7">T</font> <a href="integr22.html#K3" title="INTEGR22:func.3">.</a> <font color="Maroon" title="c9">n</font></span>)</span></span>)</span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1"><a href="normsp_0.html#K2" title="NORMSP_0:func.2">||.</a><span class="default"><font color="Maroon" title="c6">v</font></span><a href="normsp_0.html#K2" title="NORMSP_0:func.2">.||</a></span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a>  the    <a href="integr22.html#M1" title="INTEGR22:mode.1">var_volume</a> of <font color="Maroon" title="c2">rho</font>,<font color="Maroon" title="c7">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c9">n</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="rvsum_1.html#T87" onmouseover="rs('rvsum_1/T87')" onmouseout="rh()">RVSUM_1:87</a></span>;<br/>
<a NAME="E9:21_2_1"/>
<span class="p1"><a href="normsp_0.html#K2" title="NORMSP_0:func.2">||.</a><span class="default"><font color="Maroon" title="c6">v</font></span><a href="normsp_0.html#K2" title="NORMSP_0:func.2">.||</a></span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a>  the    <a href="integr22.html#M1" title="INTEGR22:mode.1">var_volume</a> of <font color="Maroon" title="c2">rho</font>,<font color="Maroon" title="c7">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c9">n</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1"><a href="normsp_0.html#K2" title="NORMSP_0:func.2">||.</a><span class="default"><font color="Maroon" title="c6">v</font></span><a href="normsp_0.html#K2" title="NORMSP_0:func.2">.||</a></span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><a href="integr22.html#K2" title="INTEGR22:func.2">total_vd</a> <font color="Maroon" title="c2">rho</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:21"><span class="lab"><font color="Green" title="E22">A1</font></span></a>, <a class="ref" href="integr22.html#T5" onmouseover="rs('integr22/T5')" onmouseout="rh()">INTEGR22:5</a>, <a class="ref" href="xreal_1.html#T64" onmouseover="rs('xreal_1/T64')" onmouseout="rh()">XREAL_1:64</a></span>;<br/>
<a NAME="E10:21_2_1"/><span class="kw">then </span>
<span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p3">(<span class="default"> the    <a href="integr22.html#M3" title="INTEGR22:mode.3">middle_volume_Sequence</a> of <font color="Maroon" title="c2">rho</font>,<font color="Maroon" title="c5">u</font>,<font color="Maroon" title="c7">T</font> <a href="integr22.html#K3" title="INTEGR22:func.3">.</a> <font color="Maroon" title="c9">n</font></span>)</span></span>)</span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1"><a href="normsp_0.html#K2" title="NORMSP_0:func.2">||.</a><span class="default"><font color="Maroon" title="c6">v</font></span><a href="normsp_0.html#K2" title="NORMSP_0:func.2">.||</a></span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><a href="integr22.html#K2" title="INTEGR22:func.2">total_vd</a> <font color="Maroon" title="c2">rho</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:21_2_1"><span class="lab"><font color="Green" title="E33">A25</font></span></a>, <a class="ref" href="xxreal_0.html#T2" onmouseover="rs('xxreal_0/T2')" onmouseout="rh()">XXREAL_0:2</a></span>;<br/>
<span class="kw">hence </span><a NAME="E11:21_2_1"/>
<span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="integr22.html#K4" title="INTEGR22:func.4">middle_sum</a>  the    <a href="integr22.html#M3" title="INTEGR22:mode.3">middle_volume_Sequence</a> of <font color="Maroon" title="c2">rho</font>,<font color="Maroon" title="c5">u</font>,<font color="Maroon" title="c7">T</font></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c9">n</font></span>)</span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1"><a href="normsp_0.html#K2" title="NORMSP_0:func.2">||.</a><span class="default"><font color="Maroon" title="c6">v</font></span><a href="normsp_0.html#K2" title="NORMSP_0:func.2">.||</a></span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><a href="integr22.html#K2" title="INTEGR22:func.2">total_vd</a> <font color="Maroon" title="c2">rho</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="integr22.html#D7" onmouseover="rs('integr22/D7')" onmouseout="rh()">INTEGR22:def 7</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">reconsider </span><font color="Maroon" title="c9">a</font> = <span class="p1"><a href="normsp_0.html#K2" title="NORMSP_0:func.2">||.</a><span class="default"><font color="Maroon" title="c6">v</font></span><a href="normsp_0.html#K2" title="NORMSP_0:func.2">.||</a></span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><a href="integr22.html#K2" title="INTEGR22:func.2">total_vd</a> <font color="Maroon" title="c2">rho</font></span>)</span> as   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a> ;<br/>
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="21_2_2"><span class="kw">now </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">  for <font color="Olive" title="b1">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> holds  <span class="p1"><a href="valued_1.html#K55" title="VALUED_1:func.55">|.</a><span class="default"><span class="p2">(<span class="default"><a href="integr22.html#K4" title="INTEGR22:func.4">middle_sum</a>  the    <a href="integr22.html#M3" title="INTEGR22:mode.3">middle_volume_Sequence</a> of <font color="Maroon" title="c2">rho</font>,<font color="Maroon" title="c5">u</font>,<font color="Maroon" title="c7">T</font></span>)</span></span><a href="valued_1.html#K55" title="VALUED_1:func.55">.|</a></span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Olive" title="b1">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="seq_1.html#K1" title="SEQ_1:func.1">seq_const</a> <font color="Maroon" title="c9">a</font></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Olive" title="b1">n</font></span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c10">n</font> be   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</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"> <span class="p1"><a href="valued_1.html#K55" title="VALUED_1:func.55">|.</a><span class="default"><span class="p2">(<span class="default"><a href="integr22.html#K4" title="INTEGR22:func.4">middle_sum</a>  the    <a href="integr22.html#M3" title="INTEGR22:mode.3">middle_volume_Sequence</a> of <font color="Maroon" title="c2">rho</font>,<font color="Maroon" title="c5">u</font>,<font color="Maroon" title="c7">T</font></span>)</span></span><a href="valued_1.html#K55" title="VALUED_1:func.55">.|</a></span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c10">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="seq_1.html#K1" title="SEQ_1:func.1">seq_const</a> <font color="Maroon" title="c9">a</font></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c10">n</font></span><br/><a NAME="E1:21_2_2"/>
<span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="integr22.html#K4" title="INTEGR22:func.4">middle_sum</a>  the    <a href="integr22.html#M3" title="INTEGR22:mode.3">middle_volume_Sequence</a> of <font color="Maroon" title="c2">rho</font>,<font color="Maroon" title="c5">u</font>,<font color="Maroon" title="c7">T</font></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c10">n</font></span>)</span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c9">a</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:21_2"><span class="lab"><font color="Green" title="E30">A10</font></span></a></span>;<br/><a NAME="E2:21_2_2"/><span class="kw">then </span>
<span class="p1"><a href="valued_1.html#K55" title="VALUED_1:func.55">|.</a><span class="default"><span class="p2">(<span class="default"><a href="integr22.html#K4" title="INTEGR22:func.4">middle_sum</a>  the    <a href="integr22.html#M3" title="INTEGR22:mode.3">middle_volume_Sequence</a> of <font color="Maroon" title="c2">rho</font>,<font color="Maroon" title="c5">u</font>,<font color="Maroon" title="c7">T</font></span>)</span></span><a href="valued_1.html#K55" title="VALUED_1:func.55">.|</a></span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c10">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c9">a</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="seq_1.html#T12" onmouseover="rs('seq_1/T12')" onmouseout="rh()">SEQ_1:12</a></span>;<br/><span class="kw">hence </span><a NAME="E3:21_2_2"/>
<span class="p1"><a href="valued_1.html#K55" title="VALUED_1:func.55">|.</a><span class="default"><span class="p2">(<span class="default"><a href="integr22.html#K4" title="INTEGR22:func.4">middle_sum</a>  the    <a href="integr22.html#M3" title="INTEGR22:mode.3">middle_volume_Sequence</a> of <font color="Maroon" title="c2">rho</font>,<font color="Maroon" title="c5">u</font>,<font color="Maroon" title="c7">T</font></span>)</span></span><a href="valued_1.html#K55" title="VALUED_1:func.55">.|</a></span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c10">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="seq_1.html#K1" title="SEQ_1:func.1">seq_const</a> <font color="Maroon" title="c9">a</font></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c10">n</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="seq_1.html#T57" onmouseover="rs('seq_1/T57')" onmouseout="rh()">SEQ_1:57</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>
<a NAME="E9:21_2"/><span class="kw">then </span>
 <a href="seq_2.html#K1" title="SEQ_2:func.1">lim</a> <span class="p1"><a href="valued_1.html#K55" title="VALUED_1:func.55">|.</a><span class="default"><span class="p2">(<span class="default"><a href="integr22.html#K4" title="INTEGR22:func.4">middle_sum</a>  the    <a href="integr22.html#M3" title="INTEGR22:mode.3">middle_volume_Sequence</a> of <font color="Maroon" title="c2">rho</font>,<font color="Maroon" title="c5">u</font>,<font color="Maroon" title="c7">T</font></span>)</span></span><a href="valued_1.html#K55" title="VALUED_1:func.55">.|</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="seq_2.html#K1" title="SEQ_2:func.1">lim</a> <span class="p1">(<span class="default"><a href="seq_1.html#K1" title="SEQ_1:func.1">seq_const</a> <font color="Maroon" title="c9">a</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:21_2"><span class="lab"><font color="Green" title="E29">A9</font></span></a>, <a class="ref" href="seq_2.html#T18" onmouseover="rs('seq_2/T18')" onmouseout="rh()">SEQ_2:18</a></span>;<br/>
<a NAME="E10:21_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E31">A27</font></span>: 
<span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><a href="seq_2.html#K1" title="SEQ_2:func.1">lim</a> <span class="p3">(<span class="default"><a href="integr22.html#K4" title="INTEGR22:func.4">middle_sum</a>  the    <a href="integr22.html#M3" title="INTEGR22:mode.3">middle_volume_Sequence</a> of <font color="Maroon" title="c2">rho</font>,<font color="Maroon" title="c5">u</font>,<font color="Maroon" title="c7">T</font></span>)</span></span>)</span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="seq_2.html#K1" title="SEQ_2:func.1">lim</a> <span class="p1">(<span class="default"><a href="seq_1.html#K1" title="SEQ_1:func.1">seq_const</a> <font color="Maroon" title="c9">a</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:21_2"><span class="lab"><font color="Green" title="E27">B6</font></span></a>, <a class="txt" href="#E4:21_2"><span class="lab"><font color="Green" title="E28">A7</font></span></a>, <a class="ref" href="seq_4.html#T14" onmouseover="rs('seq_4/T14')" onmouseout="rh()">SEQ_4:14</a></span>;<br/>
 <a href="seq_2.html#K1" title="SEQ_2:func.1">lim</a> <span class="p1">(<span class="default"><a href="seq_1.html#K1" title="SEQ_1:func.1">seq_const</a> <font color="Maroon" title="c9">a</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="seq_1.html#K1" title="SEQ_1:func.1">seq_const</a> <font color="Maroon" title="c9">a</font></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>
<span class="kw">by</span> <span class="lab"><a class="ref" href="seq_4.html#T26" onmouseover="rs('seq_4/T26')" onmouseout="rh()">SEQ_4:26</a></span>
<br/>.= 
<font color="Maroon" title="c9">a</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="seq_1.html#T57" onmouseover="rs('seq_1/T57')" onmouseout="rh()">SEQ_1:57</a></span>
;<br/>
<span class="kw">hence </span><a NAME="E12:21_2"/>
<span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><a href="integr22.html#K5" title="INTEGR22:func.5">integral</a> (<font color="Maroon" title="c5">u</font>,<font color="Maroon" title="c2">rho</font>)</span>)</span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1"><a href="normsp_0.html#K2" title="NORMSP_0:func.2">||.</a><span class="default"><font color="Maroon" title="c6">v</font></span><a href="normsp_0.html#K2" title="NORMSP_0:func.2">.||</a></span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><a href="integr22.html#K2" title="INTEGR22:func.2">total_vd</a> <font color="Maroon" title="c2">rho</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:21_2"><span class="lab"><font color="Green" title="E27">B6</font></span></a>, <a class="txt" href="#E1:21"><span class="lab"><font color="Green" title="E22">A1</font></span></a>, <a class="txt" href="#E1:21_2"><span class="lab"><font color="Green" title="E26">A5</font></span></a>, <a class="ref" href="integr22.html#D9" onmouseover="rs('integr22/D9')" onmouseout="rh()">INTEGR22:def 9</a>, <a class="txt" href="#E4:21_2"><span class="lab"><font color="Green" title="E28">A7</font></span></a>, <a class="txt" href="#E10:21_2"><span class="lab"><font color="Green" title="E31">A27</font></span></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">reconsider </span><font color="Maroon" title="c5">g</font> = <font color="Maroon" title="c3">f</font> as   <a href="dualsp01.html#V1" title="DUALSP01:attr.1">Lipschitzian</a>  <a href="hahnban.html#NM3" title="HAHNBAN:NM.3">linear-Functional</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 class="kw">by</span> <span class="lab"><a class="ref" href="dualsp01.html#D10" onmouseover="rs('dualsp01/D10')" onmouseout="rh()">DUALSP01:def 10</a></span>;<br/>
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="21_3"><span class="kw">now </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">  for <font color="Olive" title="b1">k</font> being   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a>  st <font color="Olive" title="b1">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="dualsp01.html#K15" title="DUALSP01:func.15">PreNorms</a> <font color="Maroon" title="c5">g</font> holds <br/><font color="Olive" title="b1">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="integr22.html#K2" title="INTEGR22:func.2">total_vd</a> <font color="Maroon" title="c2">rho</font></span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c6">k</font> be   <a href="xreal_0.html#NM1" title="XREAL_0:NM.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="c6">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="dualsp01.html#K15" title="DUALSP01:func.15">PreNorms</a> <font color="Maroon" title="c5">g</font> implies <font color="Maroon" title="c6">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="integr22.html#K2" title="INTEGR22:func.2">total_vd</a> <font color="Maroon" title="c2">rho</font> )</span><br/><span class="kw">assume </span><a NAME="E1:21_3"/>
<font color="Maroon" title="c6">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="dualsp01.html#K15" title="DUALSP01:func.15">PreNorms</a> <font color="Maroon" title="c5">g</font>
 ; <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="c6">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="integr22.html#K2" title="INTEGR22:func.2">total_vd</a> <font color="Maroon" title="c2">rho</font></span><br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c7">u</font> being   <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 class="kw"> such that </span><br/><a NAME="E3:21_3"/><span class="lab"><font color="Green" title="E26">A28</font></span>: 
( <font color="Maroon" title="c6">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">u</font></span>)</span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span> &amp; <span class="p1"><a href="normsp_0.html#K2" title="NORMSP_0:func.2">||.</a><span class="default"><font color="Maroon" title="c7">u</font></span><a href="normsp_0.html#K2" title="NORMSP_0:func.2">.||</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 )
 ;<br/><a NAME="E4:21_3"/>
<font color="Maroon" title="c7">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/><a NAME="E5:21_3"/><span class="kw">then </span>
 ex <font color="Olive" title="b1">v</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> st <font color="Maroon" title="c7">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">v</font>
 ;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c8">v</font> = <font color="Maroon" title="c7">u</font> as   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a> ;<br/><a NAME="E7:21_3"/><span class="lab"><font color="Green" title="E27">A29</font></span>: 
(  <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <font color="Maroon" title="c8">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">A</font> &amp; <font color="Maroon" title="c8">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="ref" href="dualsp05.html#T6" target="_self" onmouseover="rs('dualsp05/T6')" onmouseout="rh()">Th80</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c9">v</font> = <font color="Maroon" title="c8">v</font> as   <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="ref" href="dualsp05.html#T6" target="_self" onmouseover="rs('dualsp05/T6')" onmouseout="rh()">Th80</a></span>;<br/><a NAME="E9:21_3"/>
<span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><a href="integr22.html#K5" title="INTEGR22:func.5">integral</a> (<font color="Maroon" title="c9">v</font>,<font color="Maroon" title="c2">rho</font>)</span>)</span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1"><a href="normsp_0.html#K2" title="NORMSP_0:func.2">||.</a><span class="default"><font color="Maroon" title="c7">u</font></span><a href="normsp_0.html#K2" title="NORMSP_0:func.2">.||</a></span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><a href="integr22.html#K2" title="INTEGR22:func.2">total_vd</a> <font color="Maroon" title="c2">rho</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:21"><span class="lab"><font color="Green" title="E25">A4</font></span></a>, <a class="txt" href="#E7:21_3"><span class="lab"><font color="Green" title="E27">A29</font></span></a></span>;<br/><a NAME="E10:21_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E28">A30</font></span>: 
<span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">u</font></span>)</span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1"><a href="normsp_0.html#K2" title="NORMSP_0:func.2">||.</a><span class="default"><font color="Maroon" title="c7">u</font></span><a href="normsp_0.html#K2" title="NORMSP_0:func.2">.||</a></span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><a href="integr22.html#K2" title="INTEGR22:func.2">total_vd</a> <font color="Maroon" title="c2">rho</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:21"><span class="lab"><font color="Green" title="E24">A3</font></span></a></span>;<br/><a NAME="E11:21_3"/>
 <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="integr22.html#K2" title="INTEGR22:func.2">total_vd</a> <font color="Maroon" title="c2">rho</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:21"><span class="lab"><font color="Green" title="E22">A1</font></span></a>, <a class="ref" href="integr22.html#T6" onmouseover="rs('integr22/T6')" onmouseout="rh()">INTEGR22:6</a></span>;<br/><a NAME="E12:21_3"/><span class="kw">then </span>
<span class="p1"><a href="normsp_0.html#K2" title="NORMSP_0:func.2">||.</a><span class="default"><font color="Maroon" title="c7">u</font></span><a href="normsp_0.html#K2" title="NORMSP_0:func.2">.||</a></span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><a href="integr22.html#K2" title="INTEGR22:func.2">total_vd</a> <font color="Maroon" title="c2">rho</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><a href="integr22.html#K2" title="INTEGR22:func.2">total_vd</a> <font color="Maroon" title="c2">rho</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:21_3"><span class="lab"><font color="Green" title="E26">A28</font></span></a>, <a class="ref" href="xreal_1.html#T64" onmouseover="rs('xreal_1/T64')" onmouseout="rh()">XREAL_1:64</a></span>;<br/><span class="kw">hence </span><a NAME="E13:21_3"/>
<font color="Maroon" title="c6">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="integr22.html#K2" title="INTEGR22:func.2">total_vd</a> <font color="Maroon" title="c2">rho</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:21_3"><span class="lab"><font color="Green" title="E26">A28</font></span></a>, <a class="txt" href="#E10:21_3"><span class="lab"><font color="Green" title="E28">A30</font></span></a>, <a class="ref" href="xxreal_0.html#T2" onmouseover="rs('xxreal_0/T2')" onmouseout="rh()">XXREAL_0:2</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>
<a NAME="E7:21"/><span class="kw">then </span>
 <a href="seq_4.html#K2" title="SEQ_4:func.2">upper_bound</a> <span class="p1">(<span class="default"><a href="dualsp01.html#K15" title="DUALSP01:func.15">PreNorms</a> <font color="Maroon" title="c5">g</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="integr22.html#K2" title="INTEGR22:func.2">total_vd</a> <font color="Maroon" title="c2">rho</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="seq_4.html#T45" onmouseover="rs('seq_4/T45')" onmouseout="rh()">SEQ_4:45</a></span>;<br/>
<span class="kw">hence </span><a NAME="E8:21"/>
<span class="p1"><a href="normsp_0.html#K2" title="NORMSP_0:func.2">||.</a><span class="default"><font color="Maroon" title="c3">f</font></span><a href="normsp_0.html#K2" title="NORMSP_0:func.2">.||</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="integr22.html#K2" title="INTEGR22:func.2">total_vd</a> <font color="Maroon" title="c2">rho</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="dualsp01.html#T24" onmouseover="rs('dualsp01/T24')" onmouseout="rh()">DUALSP01:24</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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