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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">x</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  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="integra1.html#K3" title="INTEGRA1:func.3">vol</a> <font color="Maroon" title="c1">A</font> holds <br/> ex <font color="Olive" title="b2">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> st <br/>( <font color="Olive" title="b2">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="b1">x</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="b2">rho</font>) ) &amp; <span class="p1"><a href="normsp_0.html#K2" title="NORMSP_0:func.2">||.</a><span class="default"><font color="Olive" title="b1">x</font></span><a href="normsp_0.html#K2" title="NORMSP_0:func.2">.||</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="integr22.html#K2" title="INTEGR22:func.2">total_vd</a> <font color="Olive" title="b2">rho</font> )</span><br/><span class="kw">let </span><font color="Maroon" title="c2">x</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"> (  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="integra1.html#K3" title="INTEGRA1:func.3">vol</a> <font color="Maroon" title="c1">A</font> implies  ex <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> st <br/>( <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="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="Maroon" title="c2">x</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="Olive" title="b1">rho</font>) ) &amp; <span class="p1"><a href="normsp_0.html#K2" title="NORMSP_0:func.2">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="normsp_0.html#K2" title="NORMSP_0:func.2">.||</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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">assume </span><a NAME="E1:22"/><span class="lab"><font color="Green" title="E22">A1</font></span>: 
 <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="integra1.html#K3" title="INTEGRA1:func.3">vol</a> <font color="Maroon" title="c1">A</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">  ex <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> st <br/>( <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="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="Maroon" title="c2">x</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="Olive" title="b1">rho</font>) ) &amp; <span class="p1"><a href="normsp_0.html#K2" title="NORMSP_0:func.2">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="normsp_0.html#K2" title="NORMSP_0:func.2">.||</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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">set </span><font color="Maroon" title="c3">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/>
<span class="kw">set </span><font color="Maroon" title="c4">V</font> =  <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="p1">(<span class="default"><a href="dualsp05.html#K1" title="DUALSP05:func.1">ClstoCmp</a> <font color="Maroon" title="c1">A</font></span>)</span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c5">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="E2:22"/><span class="lab"><font color="Green" title="E23">A2</font></span>: 
 the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="dualsp05.html#K1" title="DUALSP05:func.1">ClstoCmp</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">A</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="dualsp05.html#T5" target="_self" onmouseover="rs('dualsp05/T5')" onmouseout="rh()">Lm1</a></span>;<br/>
<a NAME="E3:22"/>
 <a href="c0sp2.html#K3" title="C0SP2:func.3">R_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> is    <a href="c0sp1.html#M2" title="C0SP1:mode.2">Subalgebra</a> of  <a href="c0sp1.html#K7" title="C0SP1:func.7">R_Algebra_of_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="ref" href="c0sp2.html#T9" onmouseover="rs('c0sp2/T9')" onmouseout="rh()">C0SP2:9</a></span>;<br/>
<a NAME="E4:22"/><span class="kw">then </span><span class="lab"><font color="Green" title="E24">A3</font></span>: 
(  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="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 href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="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> &amp;  the <a href="algstr_0.html#U1" title="ALGSTR_0:sel.1">addF</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 href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the <a href="algstr_0.html#U1" title="ALGSTR_0:sel.1">addF</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> <a href="realset1.html#K1" title="REALSET1:func.1">||</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="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> &amp;  the <a href="rlvect_1.html#U1" title="RLVECT_1:sel.1">Mult</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 href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the <a href="rlvect_1.html#U1" title="RLVECT_1:sel.1">Mult</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> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>, the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <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><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="c0sp1.html#D9" onmouseover="rs('c0sp1/D9')" onmouseout="rh()">C0SP1:def 9</a></span>;<br/>
<span class="lab"><font color="Green" title="E25">A4</font></span>:  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <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="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="c0sp2.html#K1" title="C0SP2:func.1">--&gt;</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>
<span class="kw">by</span> <span class="lab"><a class="ref" href="c0sp2.html#T12" onmouseover="rs('c0sp2/T12')" onmouseout="rh()">C0SP2:12</a></span>
<br/>.= 
 <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <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="c0sp1.html#T25" onmouseover="rs('c0sp1/T25')" onmouseout="rh()">C0SP1:25</a></span>
;<br/>
<a NAME="E6:22"/><span class="lab"><font color="Green" title="E26">B5</font></span>: 
 <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> is    <a href="dualsp01.html#M1" title="DUALSP01:mode.1">SubRealNormSpace</a> of  <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="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="#E4:22"><span class="lab"><font color="Green" title="E24">A3</font></span></a>, <a class="txt" href="#E5:22"><span class="lab"><font color="Green" title="E25">A4</font></span></a>, <a class="ref" href="dualsp01.html#D16" onmouseover="rs('dualsp01/D16')" onmouseout="rh()">DUALSP01:def 16</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c6">h</font> = <font color="Maroon" title="c2">x</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/>
<span class="kw">consider </span><font color="Maroon" title="c7">f</font> being   <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="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>, <font color="Maroon" title="c8">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="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="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><span class="kw"> such that </span><br/><a NAME="E9:22"/><span class="lab"><font color="Green" title="E27">A6</font></span>: 
( <font color="Maroon" title="c7">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">F</font> &amp; <font color="Maroon" title="c7">f</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="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 href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">h</font> &amp; <span class="p1"><a href="normsp_0.html#K2" title="NORMSP_0:func.2">||.</a><span class="default"><font color="Maroon" title="c8">F</font></span><a href="normsp_0.html#K2" title="NORMSP_0:func.2">.||</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="normsp_0.html#K2" title="NORMSP_0:func.2">||.</a><span class="default"><font color="Maroon" title="c2">x</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="#E6:22"><span class="lab"><font color="Green" title="E26">B5</font></span></a>, <a class="ref" href="dualsp01.html#T36" onmouseover="rs('dualsp01/T36')" onmouseout="rh()">DUALSP01:36</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c9">a</font>, <font color="Maroon" title="c10">b</font> being   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a><span class="kw"> such that </span><br/><a NAME="E11:22"/><span class="lab"><font color="Green" title="E28">A7</font></span>: 
( <font color="Maroon" title="c9">a</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c10">b</font> &amp; <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c9">a</font>,<font color="Maroon" title="c10">b</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">A</font> &amp;  <a href="dualsp05.html#K1" title="DUALSP05:func.1">ClstoCmp</a> <font color="Maroon" title="c1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="topmetr.html#K4" title="TOPMETR:func.4">Closed-Interval-TSpace</a> (<font color="Maroon" title="c9">a</font>,<font color="Maroon" title="c10">b</font>) )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="dualsp05.html#D1" target="_self" onmouseover="rs('dualsp05/D1')" onmouseout="rh()">Def7ClstoCmp</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c11">m</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c1">A</font>,<span class="p1">(<span class="default"><a href="c0sp1.html#K6" title="C0SP1:func.6">BoundedFunctions</a> <font color="Maroon" title="c1">A</font></span>)</span><span class="kw"> such that </span><br/><a NAME="E13:22"/><span class="lab"><font color="Green" title="E29">A8</font></span>: 
 for <font color="Olive" title="b1">s</font> being   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a>  st <font color="Olive" title="b1">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c9">a</font>,<font color="Maroon" title="c10">b</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> holds <br/>( ( <font color="Olive" title="b1">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">a</font> implies <font color="Maroon" title="c11">m</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c9">a</font>,<font color="Maroon" title="c10">b</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> ) &amp; ( <font color="Olive" title="b1">s</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c9">a</font> implies <font color="Maroon" title="c11">m</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c9">a</font>,<font color="Olive" title="b1">s</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> 1</span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p1">(<span class="default"><span class="p2"><a href="rcomp_1.html#K4" title="RCOMP_1:func.4">].</a><span class="default"><font color="Olive" title="b1">s</font>,<font color="Maroon" title="c10">b</font></span><a href="rcomp_1.html#K4" title="RCOMP_1:func.4">.]</a></span> <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span>)</span> ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:22"><span class="lab"><font color="Green" title="E28">A7</font></span></a>, <a class="ref" href="dualsp05.html#T8" target="_self" onmouseover="rs('dualsp05/T8')" onmouseout="rh()">LM83</a></span>;<br/>
<a NAME="E14:22"/>
 the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</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> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="c0sp1.html#K6" title="C0SP1:func.6">BoundedFunctions</a> <font color="Maroon" title="c1">A</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="dualsp05.html#T5" target="_self" onmouseover="rs('dualsp05/T5')" onmouseout="rh()">Lm1</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c12">rho</font> = <font color="Maroon" title="c7">f</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c11">m</font> as   <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/>
<a NAME="E16:22"/><span class="lab"><font color="Green" title="E30">A9</font></span>: 
 for <font color="Olive" title="b1">D</font> being    <a href="integra1.html#M1" title="INTEGRA1:mode.1">Division</a> of <font color="Maroon" title="c1">A</font><br/>  for <font color="Olive" title="b2">K</font> being    <a href="integr22.html#M1" title="INTEGR22:mode.1">var_volume</a> of <font color="Maroon" title="c12">rho</font>,<font color="Olive" title="b1">D</font>  st <font color="Maroon" title="c9">a</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">D</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 holds <br/> <a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <font color="Olive" title="b2">K</font> <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="c2">x</font></span><a href="normsp_0.html#K2" title="NORMSP_0:func.2">.||</a></span>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="22_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c13">D</font> be    <a href="integra1.html#M1" title="INTEGRA1:mode.1">Division</a> of <font color="Maroon" title="c1">A</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">  for <font color="Olive" title="b1">K</font> being    <a href="integr22.html#M1" title="INTEGR22:mode.1">var_volume</a> of <font color="Maroon" title="c12">rho</font>,<font color="Maroon" title="c13">D</font>  st <font color="Maroon" title="c9">a</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c13">D</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 holds <br/> <a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <font color="Olive" title="b1">K</font> <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="c2">x</font></span><a href="normsp_0.html#K2" title="NORMSP_0:func.2">.||</a></span></span><br/><span class="kw">let </span><font color="Maroon" title="c14">K</font> be    <a href="integr22.html#M1" title="INTEGR22:mode.1">var_volume</a> of <font color="Maroon" title="c12">rho</font>,<font color="Maroon" title="c13">D</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="c9">a</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c13">D</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 implies  <a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <font color="Maroon" title="c14">K</font> <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="c2">x</font></span><a href="normsp_0.html#K2" title="NORMSP_0:func.2">.||</a></span> )</span><br/>



<span class="kw">assume </span><a NAME="E1:22_2"/><span class="lab"><font color="Green" title="E31">A10</font></span>: 
<font color="Maroon" title="c9">a</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c13">D</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1
 ; <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="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <font color="Maroon" title="c14">K</font> <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="c2">x</font></span><a href="normsp_0.html#K2" title="NORMSP_0:func.2">.||</a></span></span><br/>

<span class="kw">consider </span><font color="Maroon" title="c15">s</font> being   <a href="struct_0.html#NM7" title="STRUCT_0:NM.7">FinSequence</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"> such that </span><br/><a NAME="E3:22_2"/><span class="lab"><font color="Green" title="E32">A11</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c15">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c13">D</font>
 <span class="kw">and </span><br/><a NAME="E4:22_2"/><span class="lab"><font color="Green" title="E33">A12</font></span>: 
 for <font color="Olive" title="b1">i</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Olive" title="b1">i</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> <font color="Maroon" title="c15">s</font> holds <br/><font color="Maroon" title="c15">s</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="absvalue.html#K1" title="ABSVALUE:func.1">sgn</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="dualsp05.html#K3" title="DUALSP05:func.3">Dp2</a> (<font color="Maroon" title="c12">rho</font>,<font color="Maroon" title="c13">D</font>,<span class="p4">(<span class="default"><font color="Olive" title="b1">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>)</span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p3">(<span class="default"><a href="dualsp05.html#K3" title="DUALSP05:func.3">Dp2</a> (<font color="Maroon" title="c12">rho</font>,<font color="Maroon" title="c13">D</font>,<font color="Olive" title="b1">i</font>)</span>)</span></span>)</span></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="dualsp05.html#K2" title="DUALSP05:func.2">Dp1</a> (<font color="Maroon" title="c11">m</font>,<font color="Maroon" title="c13">D</font>,<span class="p3">(<span class="default"><font color="Olive" title="b1">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>)</span>)</span> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <span class="p2">(<span class="default"><a href="dualsp05.html#K2" title="DUALSP05:func.2">Dp1</a> (<font color="Maroon" title="c11">m</font>,<font color="Maroon" title="c13">D</font>,<font color="Olive" title="b1">i</font>)</span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="dualsp05.html#T9" target="_self" onmouseover="rs('dualsp05/T9')" onmouseout="rh()">LM84</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c16">yD</font> =  <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">Sum</a> <font color="Maroon" title="c15">s</font>;<br/>
<a NAME="E5:22_2"/>
 <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">Sum</a> <font color="Maroon" title="c15">s</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="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>
 
;<br/>
<a NAME="E6:22_2"/><span class="kw">then </span>
 <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">Sum</a> <font color="Maroon" title="c15">s</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="c0sp1.html#K6" title="C0SP1:func.6">BoundedFunctions</a> <font color="Maroon" title="c1">A</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="dualsp05.html#T5" target="_self" onmouseover="rs('dualsp05/T5')" onmouseout="rh()">Lm1</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c17">g</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><span class="kw"> such that </span><br/><a NAME="E8:22_2"/><span class="lab"><font color="Green" title="E34">A13</font></span>: 
( <font color="Maroon" title="c17">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">Sum</a> <font color="Maroon" title="c15">s</font> &amp; <font color="Maroon" title="c17">g</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c1">A</font> is  <a href="comseq_2.html#V1" title="COMSEQ_2:attr.1">bounded</a>  )
 ;<br/>
<a NAME="E9:22_2"/><span class="lab"><font color="Green" title="E35">A14</font></span>: 
<font color="Maroon" title="c7">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K4" title="RLVECT_1:func.4">Sum</a> <font color="Maroon" title="c15">s</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">f</font> <a href="finseqop.html#K4" title="FINSEQOP:func.4">*</a> <font color="Maroon" title="c15">s</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="dualsp05.html#T10" target="_self" onmouseover="rs('dualsp05/T10')" onmouseout="rh()">LM87</a></span>;<br/>
<a NAME="E10:22_2"/><span class="lab"><font color="Green" title="E36">A15</font></span>: 
 for <font color="Olive" title="b1">t</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c1">A</font> holds  <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">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">t</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> 1
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="22_2_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c18">t</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c1">A</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"><font color="Maroon" title="c17">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c18">t</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> 1</span><br/>

<span class="kw">defpred </span><font color="Maroon">S<sub>1</sub></font>[  <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>,   <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a> ] <span class="kw">means </span> ex <font color="Olive" title="b1">sk</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> st <br/>( <font color="Olive" title="b1">sk</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">s</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> $1 &amp; $2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">sk</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c18">t</font> );<br/>
<a NAME="E1:22_2_1"/><span class="lab"><font color="Green" title="E37">A16</font></span>: 
 for <font color="Olive" title="b1">k</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Olive" title="b1">k</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> <font color="Maroon" title="c13">D</font></span>)</span> holds <br/> ex <font color="Olive" title="b2">x</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">k</font>,<font color="Olive" title="b2">x</font>]
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="22_2_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c19">k</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="c19">k</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> <font color="Maroon" title="c13">D</font></span>)</span> implies  ex <font color="Olive" title="b1">x</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c19">k</font>,<font color="Olive" title="b1">x</font>] )</span><br/>

<span class="kw">assume </span><a NAME="E1:22_2_1_1"/>
<font color="Maroon" title="c19">k</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> <font color="Maroon" title="c13">D</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">  ex <font color="Olive" title="b1">x</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c19">k</font>,<font color="Olive" title="b1">x</font>]</span><br/>

<a NAME="E2:22_2_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c19">k</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> <font color="Maroon" title="c15">s</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>, <a class="txt" href="#E3:22_2"><span class="lab"><font color="Green" title="E32">A11</font></span></a></span>;<br/>
<a NAME="E3:22_2_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c15">s</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="absvalue.html#K1" title="ABSVALUE:func.1">sgn</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="dualsp05.html#K3" title="DUALSP05:func.3">Dp2</a> (<font color="Maroon" title="c12">rho</font>,<font color="Maroon" title="c13">D</font>,<span class="p4">(<span class="default"><font color="Maroon" title="c19">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>)</span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p3">(<span class="default"><a href="dualsp05.html#K3" title="DUALSP05:func.3">Dp2</a> (<font color="Maroon" title="c12">rho</font>,<font color="Maroon" title="c13">D</font>,<font color="Maroon" title="c19">k</font>)</span>)</span></span>)</span></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="dualsp05.html#K2" title="DUALSP05:func.2">Dp1</a> (<font color="Maroon" title="c11">m</font>,<font color="Maroon" title="c13">D</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c19">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>)</span>)</span> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <span class="p2">(<span class="default"><a href="dualsp05.html#K2" title="DUALSP05:func.2">Dp1</a> (<font color="Maroon" title="c11">m</font>,<font color="Maroon" title="c13">D</font>,<font color="Maroon" title="c19">k</font>)</span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:22_2"><span class="lab"><font color="Green" title="E33">A12</font></span></a></span>;<br/>
<a NAME="E4:22_2_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c15">s</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">k</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="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>
 
;<br/>
<a NAME="E5:22_2_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c15">s</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">k</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="c0sp1.html#K6" title="C0SP1:func.6">BoundedFunctions</a> <font color="Maroon" title="c1">A</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="dualsp05.html#T5" target="_self" onmouseover="rs('dualsp05/T5')" onmouseout="rh()">Lm1</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c20">sk</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><span class="kw"> such that </span><br/><a NAME="E7:22_2_1_1"/><span class="lab"><font color="Green" title="E38">A17</font></span>: 
( <font color="Maroon" title="c20">sk</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">s</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">k</font> &amp; <font color="Maroon" title="c20">sk</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c1">A</font> is  <a href="comseq_2.html#V1" title="COMSEQ_2:attr.1">bounded</a>  )
 ;<br/>
<span class="kw">take </span><font color="Maroon" title="c21">x</font> = <font color="Maroon" title="c20">sk</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c18">t</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">S<sub>1</sub></font>[<font color="Maroon" title="c19">k</font>,<font color="Maroon" title="c21">x</font>]</span><br/>

<span class="kw">thus </span><a NAME="E8:22_2_1_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c19">k</font>,<font color="Maroon" title="c21">x</font>]
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:22_2_1_1"><span class="lab"><font color="Green" title="E38">A17</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">consider </span><font color="Maroon" title="c19">z</font> being    <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> <span class="kw"> such that </span><br/><a NAME="E3:22_2_1"/><span class="lab"><font color="Green" title="E38">A18</font></span>: 
(  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c19">z</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> <font color="Maroon" title="c13">D</font></span>)</span> &amp; (  for <font color="Olive" title="b1">k</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Olive" title="b1">k</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> <font color="Maroon" title="c13">D</font></span>)</span> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">k</font>,<font color="Maroon" title="c19">z</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">k</font>] ) )
 <span class="kw">from</span> <span class="lab"><a class="ref" href="finseq_1.html#S5" onmouseover="rs('finseq_1/S5')" onmouseout="rh()">FINSEQ_1:sch 5</a>(<a class="txt" href="#E1:22_2_1"><span class="lab"><font color="Green" title="E37">A16</font></span></a>);<br/></span>
<a NAME="E4:22_2_1"/><span class="lab"><font color="Green" title="E39">A20</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c15">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c19">z</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:22_2"><span class="lab"><font color="Green" title="E32">A11</font></span></a>, <a class="txt" href="#E3:22_2_1"><span class="lab"><font color="Green" title="E38">A18</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="E5:22_2_1"/><span class="lab"><font color="Green" title="E40">A22</font></span>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c19">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c13">D</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>, <a class="txt" href="#E3:22_2_1"><span class="lab"><font color="Green" title="E38">A18</font></span></a></span>;<br/>
<a NAME="E6:22_2_1"/>
<font color="Maroon" title="c1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><span class="p2">(<span class="default"><a href="seq_4.html#K3" title="SEQ_4:func.3">lower_bound</a> <font color="Maroon" title="c1">A</font></span>)</span>,<span class="p2">(<span class="default"><a href="seq_4.html#K2" title="SEQ_4:func.2">upper_bound</a> <font color="Maroon" title="c1">A</font></span>)</span></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="integra1.html#T4" onmouseover="rs('integra1/T4')" onmouseout="rh()">INTEGRA1:4</a></span>;<br/>
<a NAME="E7:22_2_1"/><span class="kw">then </span>
 <a href="seq_4.html#K3" title="SEQ_4:func.3">lower_bound</a> <font color="Maroon" title="c1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">a</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:22"><span class="lab"><font color="Green" title="E28">A7</font></span></a>, <a class="ref" href="integra1.html#T5" onmouseover="rs('integra1/T5')" onmouseout="rh()">INTEGRA1:5</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c20">i</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a> <span class="kw"> such that </span><br/><a NAME="E9:22_2_1"/><span class="lab"><font color="Green" title="E41">A50</font></span>: 
<font color="Maroon" title="c20">i</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> <font color="Maroon" title="c13">D</font>
 <span class="kw">and </span><br/><a NAME="E10:22_2_1"/><span class="lab"><font color="Green" title="E42">A51</font></span>: 
<font color="Maroon" title="c18">t</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c13">D</font>,<font color="Maroon" title="c20">i</font>)
 <span class="kw">and </span><br/><a NAME="E11:22_2_1"/><span class="lab"><font color="Green" title="E43">A52</font></span>: 
( <font color="Maroon" title="c20">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or (  <a href="seq_4.html#K3" title="SEQ_4:func.3">lower_bound</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c13">D</font>,<font color="Maroon" title="c20">i</font>)</span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c18">t</font> &amp; <font color="Maroon" title="c18">t</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="seq_4.html#K2" title="SEQ_4:func.2">upper_bound</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c13">D</font>,<font color="Maroon" title="c20">i</font>)</span>)</span> ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:22_2"><span class="lab"><font color="Green" title="E31">A10</font></span></a>, <a class="ref" href="dualsp05.html#T13" target="_self" onmouseover="rs('dualsp05/T13')" onmouseout="rh()">LM94</a></span>;<br/>
<a NAME="E12:22_2_1"/>
<font color="Maroon" title="c18">t</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><span class="p2">(<span class="default"><a href="seq_4.html#K3" title="SEQ_4:func.3">lower_bound</a> <span class="p3">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c13">D</font>,<font color="Maroon" title="c20">i</font>)</span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="seq_4.html#K2" title="SEQ_4:func.2">upper_bound</a> <span class="p3">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c13">D</font>,<font color="Maroon" title="c20">i</font>)</span>)</span></span>)</span></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:22_2_1"><span class="lab"><font color="Green" title="E42">A51</font></span></a>, <a class="ref" href="integra1.html#T4" onmouseover="rs('integra1/T4')" onmouseout="rh()">INTEGRA1:4</a></span>;<br/>
<a NAME="E13:22_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E44">A53</font></span>: 
(  <a href="seq_4.html#K3" title="SEQ_4:func.3">lower_bound</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c13">D</font>,<font color="Maroon" title="c20">i</font>)</span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c18">t</font> &amp; <font color="Maroon" title="c18">t</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="seq_4.html#K2" title="SEQ_4:func.2">upper_bound</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c13">D</font>,<font color="Maroon" title="c20">i</font>)</span>)</span> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_1.html#T1" onmouseover="rs('xxreal_1/T1')" onmouseout="rh()">XXREAL_1:1</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c21">i</font> = <font color="Maroon" title="c20">i</font> as   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> ;<br/>
<a NAME="E15:22_2_1"/><span class="lab"><font color="Green" title="E45">A54</font></span>: 
<font color="Maroon" title="c21">i</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> <font color="Maroon" title="c13">D</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:22_2_1"><span class="lab"><font color="Green" title="E41">A50</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/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c22">si</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><span class="kw"> such that </span><br/><a NAME="E17:22_2_1"/><span class="lab"><font color="Green" title="E46">A55</font></span>: 
( <font color="Maroon" title="c22">si</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">s</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">i</font> &amp; <font color="Maroon" title="c19">z</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">si</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c18">t</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:22_2_1"><span class="lab"><font color="Green" title="E38">A18</font></span></a></span>;<br/>
<a NAME="E18:22_2_1"/>
<font color="Maroon" title="c21">i</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> <font color="Maroon" title="c15">s</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:22_2"><span class="lab"><font color="Green" title="E32">A11</font></span></a>, <a class="txt" href="#E9:22_2_1"><span class="lab"><font color="Green" title="E41">A50</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="E19:22_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E47">B56</font></span>: 
<font color="Maroon" title="c21">i</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> <font color="Maroon" title="c15">s</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/>
<span class="kw">set </span><font color="Maroon" title="c23">r</font> =  <a href="absvalue.html#K1" title="ABSVALUE:func.1">sgn</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="dualsp05.html#K3" title="DUALSP05:func.3">Dp2</a> (<font color="Maroon" title="c12">rho</font>,<font color="Maroon" title="c13">D</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c21">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>)</span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p2">(<span class="default"><a href="dualsp05.html#K3" title="DUALSP05:func.3">Dp2</a> (<font color="Maroon" title="c12">rho</font>,<font color="Maroon" title="c13">D</font>,<font color="Maroon" title="c21">i</font>)</span>)</span></span>)</span>;<br/>
<a NAME="E20:22_2_1"/><span class="lab"><font color="Green" title="E48">A57</font></span>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c21">i</font> &amp; <font color="Maroon" title="c21">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c13">D</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:22_2_1"><span class="lab"><font color="Green" title="E41">A50</font></span></a>, <a class="ref" href="finseq_3.html#T25" onmouseover="rs('finseq_3/T25')" onmouseout="rh()">FINSEQ_3:25</a></span>;<br/>
<a NAME="E21:22_2_1"/><span class="kw">then </span>
<font color="Maroon" title="c21">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c13">D</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T7" onmouseover="rs('xreal_1/T7')" onmouseout="rh()">XREAL_1:7</a></span>;<br/>
<a NAME="E22:22_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E49">A58</font></span>: 
<font color="Maroon" title="c21">i</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"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c13">D</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:22_2_1"><span class="lab"><font color="Green" title="E48">A57</font></span></a></span>;<br/>
<a NAME="E23:22_2_1"/><span class="lab"><font color="Green" title="E50">b</font></span>: 
( 1 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c21">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <font color="Maroon" title="c21">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c13">D</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:22_2_1"><span class="lab"><font color="Green" title="E45">A54</font></span></a>, <a class="ref" href="finseq_1.html#T1" onmouseover="rs('finseq_1/T1')" onmouseout="rh()">FINSEQ_1:1</a>, <a class="ref" href="xreal_1.html#T7" onmouseover="rs('xreal_1/T7')" onmouseout="rh()">XREAL_1:7</a></span>;<br/>
<a NAME="E24:22_2_1"/><span class="kw">then </span>
<font color="Maroon" title="c21">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c13">D</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T7" onmouseover="rs('xreal_1/T7')" onmouseout="rh()">XREAL_1:7</a></span>;<br/>
<a NAME="E25:22_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E51">A59</font></span>: 
<font color="Maroon" title="c21">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <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"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c13">D</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E23:22_2_1"><span class="lab"><font color="Green" title="E50">b</font></span></a></span>;<br/>
<a NAME="E26:22_2_1"/>
<font color="Maroon" title="c19">z</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="absvalue.html#K1" title="ABSVALUE:func.1">sgn</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="dualsp05.html#K3" title="DUALSP05:func.3">Dp2</a> (<font color="Maroon" title="c12">rho</font>,<font color="Maroon" title="c13">D</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c21">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>)</span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p2">(<span class="default"><a href="dualsp05.html#K3" title="DUALSP05:func.3">Dp2</a> (<font color="Maroon" title="c12">rho</font>,<font color="Maroon" title="c13">D</font>,<font color="Maroon" title="c21">i</font>)</span>)</span></span>)</span>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="22_2_1_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">set </span><font color="Maroon" title="c24">f0</font> = <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c9">a</font>,<font color="Maroon" title="c10">b</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>;<br/>
<span class="kw">set </span><font color="Maroon" title="c25">f1</font> = <span class="p1">(<span class="default"><span class="p2"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c9">a</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c13">D</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">i</font></span>)</span></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> 1</span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p1">(<span class="default"><span class="p2"><a href="rcomp_1.html#K4" title="RCOMP_1:func.4">].</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c13">D</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">i</font></span>)</span>,<font color="Maroon" title="c10">b</font></span><a href="rcomp_1.html#K4" title="RCOMP_1:func.4">.]</a></span> <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span>)</span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c26">g1</font> = <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c9">a</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c13">D</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">i</font></span>)</span></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> 1;<br/>
<span class="kw">set </span><font color="Maroon" title="c27">g2</font> = <span class="p1"><a href="rcomp_1.html#K4" title="RCOMP_1:func.4">].</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c13">D</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">i</font></span>)</span>,<font color="Maroon" title="c10">b</font></span><a href="rcomp_1.html#K4" title="RCOMP_1:func.4">.]</a></span> <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>;<br/>
<span class="kw">set </span><font color="Maroon" title="c28">f2</font> = <span class="p1">(<span class="default"><span class="p2"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c9">a</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c13">D</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c21">i</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span></span>)</span></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> 1</span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p1">(<span class="default"><span class="p2"><a href="rcomp_1.html#K4" title="RCOMP_1:func.4">].</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c13">D</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c21">i</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c10">b</font></span><a href="rcomp_1.html#K4" title="RCOMP_1:func.4">.]</a></span> <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span>)</span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c29">h1</font> = <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c9">a</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c13">D</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c21">i</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span></span>)</span></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> 1;<br/>
<span class="kw">set </span><font color="Maroon" title="c30">h2</font> = <span class="p1"><a href="rcomp_1.html#K4" title="RCOMP_1:func.4">].</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c13">D</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c21">i</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c10">b</font></span><a href="rcomp_1.html#K4" title="RCOMP_1:func.4">.]</a></span> <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>;<br/>
<a NAME="E1:22_2_1_2"/><span class="lab"><font color="Green" title="E52">B0</font></span>: 
(  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c9">a</font>,<font color="Maroon" title="c10">b</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c9">a</font>,<font color="Maroon" title="c10">b</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> &amp;  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c9">a</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c13">D</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">i</font></span>)</span></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c9">a</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c13">D</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">i</font></span>)</span></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> &amp;  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2"><a href="rcomp_1.html#K4" title="RCOMP_1:func.4">].</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c13">D</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">i</font></span>)</span>,<font color="Maroon" title="c10">b</font></span><a href="rcomp_1.html#K4" title="RCOMP_1:func.4">.]</a></span> <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="rcomp_1.html#K4" title="RCOMP_1:func.4">].</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c13">D</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">i</font></span>)</span>,<font color="Maroon" title="c10">b</font></span><a href="rcomp_1.html#K4" title="RCOMP_1:func.4">.]</a></span> &amp;  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c9">a</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c13">D</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c21">i</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span></span>)</span></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c9">a</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c13">D</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c21">i</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span></span>)</span></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> &amp;  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2"><a href="rcomp_1.html#K4" title="RCOMP_1:func.4">].</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c13">D</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c21">i</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c10">b</font></span><a href="rcomp_1.html#K4" title="RCOMP_1:func.4">.]</a></span> <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="rcomp_1.html#K4" title="RCOMP_1:func.4">].</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c13">D</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c21">i</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c10">b</font></span><a href="rcomp_1.html#K4" title="RCOMP_1:func.4">.]</a></span> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funcop_1.html#T13" onmouseover="rs('funcop_1/T13')" onmouseout="rh()">FUNCOP_1:13</a></span>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="22_2_1_2_1"><span class="kw">per </span></a><a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:22_2_1_2_1"/>
( <font color="Maroon" title="c21">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or <font color="Maroon" title="c21">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 1 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="22_2_1_2_1_1"><span class="kw">suppose </span></a><a NAME="E1:22_2_1_2_1_1"/><span class="lab"><font color="Green" title="E53">A62</font></span>: 
<font color="Maroon" title="c21">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 ; <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="c19">z</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="absvalue.html#K1" title="ABSVALUE:func.1">sgn</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="dualsp05.html#K3" title="DUALSP05:func.3">Dp2</a> (<font color="Maroon" title="c12">rho</font>,<font color="Maroon" title="c13">D</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c21">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>)</span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p2">(<span class="default"><a href="dualsp05.html#K3" title="DUALSP05:func.3">Dp2</a> (<font color="Maroon" title="c12">rho</font>,<font color="Maroon" title="c13">D</font>,<font color="Maroon" title="c21">i</font>)</span>)</span></span>)</span></span><br/><div class="add"><a NAME="E2:22_2_1_2_1_1"/>
<font color="Maroon" title="c1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><span class="p2">(<span class="default"><a href="seq_4.html#K3" title="SEQ_4:func.3">lower_bound</a> <font color="Maroon" title="c1">A</font></span>)</span>,<span class="p2">(<span class="default"><a href="seq_4.html#K2" title="SEQ_4:func.2">upper_bound</a> <font color="Maroon" title="c1">A</font></span>)</span></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="integra1.html#T4" onmouseover="rs('integra1/T4')" onmouseout="rh()">INTEGRA1:4</a></span>;<br/><a NAME="E3:22_2_1_2_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E54">A63</font></span>: 
 <a href="seq_4.html#K3" title="SEQ_4:func.3">lower_bound</a> <font color="Maroon" title="c1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">a</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:22"><span class="lab"><font color="Green" title="E28">A7</font></span></a>, <a class="ref" href="integra1.html#T5" onmouseover="rs('integra1/T5')" onmouseout="rh()">INTEGRA1:5</a></span>;<br/><a NAME="E4:22_2_1_2_1_1"/><span class="lab"><font color="Green" title="E55">A64</font></span>: 
<font color="Maroon" title="c9">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c9">a</font>,<font color="Maroon" title="c10">b</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:22"><span class="lab"><font color="Green" title="E28">A7</font></span></a></span>;<br/><a NAME="E5:22_2_1_2_1_1"/><span class="lab"><font color="Green" title="E56">A65</font></span>: 
<font color="Maroon" title="c13">D</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">i</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c9">a</font>,<font color="Maroon" title="c10">b</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:22"><span class="lab"><font color="Green" title="E28">A7</font></span></a>, <a class="txt" href="#E9:22_2_1"><span class="lab"><font color="Green" title="E41">A50</font></span></a>, <a class="ref" href="integra1.html#T6" onmouseover="rs('integra1/T6')" onmouseout="rh()">INTEGRA1:6</a></span>;<br/><a NAME="E6:22_2_1_2_1_1"/><span class="lab"><font color="Green" title="E57">A66</font></span>: 
(  <a href="seq_4.html#K3" title="SEQ_4:func.3">lower_bound</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c13">D</font>,<font color="Maroon" title="c21">i</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="seq_4.html#K3" title="SEQ_4:func.3">lower_bound</a> <font color="Maroon" title="c1">A</font> &amp;  <a href="seq_4.html#K2" title="SEQ_4:func.2">upper_bound</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c13">D</font>,<font color="Maroon" title="c21">i</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">D</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">i</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:22_2_1"><span class="lab"><font color="Green" title="E41">A50</font></span></a>, <a class="txt" href="#E1:22_2_1_2_1_1"><span class="lab"><font color="Green" title="E53">A62</font></span></a>, <a class="ref" href="integra1.html#D4" onmouseover="rs('integra1/D4')" onmouseout="rh()">INTEGRA1:def 4</a></span>;<br/><span class="lab"><font color="Green" title="E58">A69</font></span>:  <a href="dualsp05.html#K2" title="DUALSP05:func.2">Dp1</a> (<font color="Maroon" title="c11">m</font>,<font color="Maroon" title="c13">D</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c21">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>) = 
<font color="Maroon" title="c11">m</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">D</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c21">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E25:22_2_1"><span class="lab"><font color="Green" title="E51">A59</font></span></a>, <a class="ref" href="dualsp05.html#D2" target="_self" onmouseover="rs('dualsp05/D2')" onmouseout="rh()">defDp1</a>, <a class="txt" href="#E1:22_2_1_2_1_1"><span class="lab"><font color="Green" title="E53">A62</font></span></a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c9">a</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c13">D</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">i</font></span>)</span></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> 1</span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p1">(<span class="default"><span class="p2"><a href="rcomp_1.html#K4" title="RCOMP_1:func.4">].</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c13">D</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">i</font></span>)</span>,<font color="Maroon" title="c10">b</font></span><a href="rcomp_1.html#K4" title="RCOMP_1:func.4">.]</a></span> <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:22"><span class="lab"><font color="Green" title="E29">A8</font></span></a>, <a class="txt" href="#E5:22_2_1_2_1_1"><span class="lab"><font color="Green" title="E56">A65</font></span></a>, <a class="txt" href="#E1:22_2"><span class="lab"><font color="Green" title="E31">A10</font></span></a>, <a class="txt" href="#E1:22_2_1_2_1_1"><span class="lab"><font color="Green" title="E53">A62</font></span></a></span>
;<br/><span class="lab"><font color="Green" title="E59">A70</font></span>:  <a href="dualsp05.html#K2" title="DUALSP05:func.2">Dp1</a> (<font color="Maroon" title="c11">m</font>,<font color="Maroon" title="c13">D</font>,<font color="Maroon" title="c21">i</font>) = 
<font color="Maroon" title="c11">m</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="seq_4.html#K3" title="SEQ_4:func.3">lower_bound</a> <font color="Maroon" title="c1">A</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E22:22_2_1"><span class="lab"><font color="Green" title="E49">A58</font></span></a>, <a class="txt" href="#E1:22_2_1_2_1_1"><span class="lab"><font color="Green" title="E53">A62</font></span></a>, <a class="ref" href="dualsp05.html#D2" target="_self" onmouseover="rs('dualsp05/D2')" onmouseout="rh()">defDp1</a></span>
<br/>.= 
<span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c9">a</font>,<font color="Maroon" title="c10">b</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:22"><span class="lab"><font color="Green" title="E29">A8</font></span></a>, <a class="txt" href="#E3:22_2_1_2_1_1"><span class="lab"><font color="Green" title="E54">A63</font></span></a>, <a class="txt" href="#E4:22_2_1_2_1_1"><span class="lab"><font color="Green" title="E55">A64</font></span></a></span>
;<br/><a NAME="E9:22_2_1_2_1_1"/><span class="lab"><font color="Green" title="E60">A72</font></span>: 
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c9">a</font>,<font color="Maroon" title="c10">b</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">A</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:22"><span class="lab"><font color="Green" title="E28">A7</font></span></a>, <a class="ref" href="funcop_1.html#T13" onmouseover="rs('funcop_1/T13')" onmouseout="rh()">FUNCOP_1:13</a></span>;<br/><a NAME="E10:22_2_1_2_1_1"/><span class="lab"><font color="Green" title="E61">A73</font></span>: 
( <font color="Maroon" title="c9">a</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c13">D</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">i</font> &amp; <font color="Maroon" title="c13">D</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c10">b</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:22_2_1_2_1_1"><span class="lab"><font color="Green" title="E56">A65</font></span></a>, <a class="ref" href="xxreal_1.html#T1" onmouseover="rs('xxreal_1/T1')" onmouseout="rh()">XXREAL_1:1</a></span>;<br/><span class="lab"><font color="Green" title="E62">A74</font></span>:  <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c9">a</font>,<span class="p4">(<span class="default"><font color="Maroon" title="c13">D</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">i</font></span>)</span></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> 1</span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p2">(<span class="default"><span class="p3"><a href="rcomp_1.html#K4" title="RCOMP_1:func.4">].</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c13">D</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">i</font></span>)</span>,<font color="Maroon" title="c10">b</font></span><a href="rcomp_1.html#K4" title="RCOMP_1:func.4">.]</a></span> <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p2">(<span class="default"><span class="p3"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c9">a</font>,<span class="p4">(<span class="default"><font color="Maroon" title="c13">D</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">i</font></span>)</span></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> 1</span>)</span></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p2">(<span class="default"><span class="p3"><a href="rcomp_1.html#K4" title="RCOMP_1:func.4">].</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c13">D</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">i</font></span>)</span>,<font color="Maroon" title="c10">b</font></span><a href="rcomp_1.html#K4" title="RCOMP_1:func.4">.]</a></span> <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_4.html#D1" onmouseover="rs('funct_4/D1')" onmouseout="rh()">FUNCT_4:def 1</a></span>
<br/>.= 
<font color="Maroon" title="c1">A</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:22"><span class="lab"><font color="Green" title="E28">A7</font></span></a>, <a class="txt" href="#E1:22_2_1_2"><span class="lab"><font color="Green" title="E52">B0</font></span></a>, <a class="txt" href="#E10:22_2_1_2_1_1"><span class="lab"><font color="Green" title="E61">A73</font></span></a>, <a class="ref" href="xxreal_1.html#T167" onmouseover="rs('xxreal_1/T167')" onmouseout="rh()">XXREAL_1:167</a></span>
;<br/><a NAME="E12:22_2_1_2_1_1"/>
 <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <span class="p1">(<span class="default"><span class="p2"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c9">a</font>,<font color="Maroon" title="c10">b</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> 
 ;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c31">f0</font> = <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c9">a</font>,<font color="Maroon" title="c10">b</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> as   <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> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:22_2_1_2_1_1"><span class="lab"><font color="Green" title="E60">A72</font></span></a>, <a class="ref" href="funct_2.html#T2" onmouseover="rs('funct_2/T2')" onmouseout="rh()">FUNCT_2:2</a></span>;<br/><a NAME="E14:22_2_1_2_1_1"/>
 <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c9">a</font>,<span class="p4">(<span class="default"><font color="Maroon" title="c13">D</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">i</font></span>)</span></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> 1</span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p2">(<span class="default"><span class="p3"><a href="rcomp_1.html#K4" title="RCOMP_1:func.4">].</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c13">D</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">i</font></span>)</span>,<font color="Maroon" title="c10">b</font></span><a href="rcomp_1.html#K4" title="RCOMP_1:func.4">.]</a></span> <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span>)</span></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> 
 ;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c32">f1</font> = <span class="p1">(<span class="default"><span class="p2"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c9">a</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c13">D</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">i</font></span>)</span></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> 1</span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p1">(<span class="default"><span class="p2"><a href="rcomp_1.html#K4" title="RCOMP_1:func.4">].</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c13">D</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">i</font></span>)</span>,<font color="Maroon" title="c10">b</font></span><a href="rcomp_1.html#K4" title="RCOMP_1:func.4">.]</a></span> <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span>)</span> as   <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> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:22_2_1_2_1_1"><span class="lab"><font color="Green" title="E62">A74</font></span></a>, <a class="ref" href="funct_2.html#T2" onmouseover="rs('funct_2/T2')" onmouseout="rh()">FUNCT_2:2</a></span>;<br/><a NAME="E16:22_2_1_2_1_1"/><span class="lab"><font color="Green" title="E63">A76</font></span>: 
<font color="Maroon" title="c22">si</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c18">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="absvalue.html#K1" title="ABSVALUE:func.1">sgn</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="dualsp05.html#K3" title="DUALSP05:func.3">Dp2</a> (<font color="Maroon" title="c12">rho</font>,<font color="Maroon" title="c13">D</font>,<span class="p4">(<span class="default"><font color="Maroon" title="c21">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>)</span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p3">(<span class="default"><a href="dualsp05.html#K3" title="DUALSP05:func.3">Dp2</a> (<font color="Maroon" title="c12">rho</font>,<font color="Maroon" title="c13">D</font>,<font color="Maroon" title="c21">i</font>)</span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c32">f1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c18">t</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c31">f0</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c18">t</font></span>)</span></span>)</span>
 <div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="22_2_1_2_1_1_4"><span class="kw">proof </span></a><div class="add">

<span class="kw">reconsider </span><font color="Maroon" title="c33">H</font> = <span class="p1">(<span class="default"><a href="dualsp05.html#K2" title="DUALSP05:func.2">Dp1</a> (<font color="Maroon" title="c11">m</font>,<font color="Maroon" title="c13">D</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c21">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>)</span>)</span> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <span class="p1">(<span class="default"><a href="dualsp05.html#K2" title="DUALSP05:func.2">Dp1</a> (<font color="Maroon" title="c11">m</font>,<font color="Maroon" title="c13">D</font>,<font color="Maroon" title="c21">i</font>)</span>)</span> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="c0sp1.html#K11" title="C0SP1:func.11">R_Normed_Algebra_of_BoundedFunctions</a> <font color="Maroon" title="c1">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="c34">h</font> = <font color="Maroon" title="c33">H</font> as   <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> <span class="kw">by</span> <span class="lab"><a class="ref" href="dualsp05.html#T11" target="_self" onmouseover="rs('dualsp05/T11')" onmouseout="rh()">LM88</a></span>;<br/>
<a NAME="E3:22_2_1_2_1_1_4"/>
<font color="Maroon" title="c22">si</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="absvalue.html#K1" title="ABSVALUE:func.1">sgn</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="dualsp05.html#K3" title="DUALSP05:func.3">Dp2</a> (<font color="Maroon" title="c12">rho</font>,<font color="Maroon" title="c13">D</font>,<span class="p4">(<span class="default"><font color="Maroon" title="c21">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>)</span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p3">(<span class="default"><a href="dualsp05.html#K3" title="DUALSP05:func.3">Dp2</a> (<font color="Maroon" title="c12">rho</font>,<font color="Maroon" title="c13">D</font>,<font color="Maroon" title="c21">i</font>)</span>)</span></span>)</span></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="dualsp05.html#K2" title="DUALSP05:func.2">Dp1</a> (<font color="Maroon" title="c11">m</font>,<font color="Maroon" title="c13">D</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c21">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>)</span>)</span> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <span class="p2">(<span class="default"><a href="dualsp05.html#K2" title="DUALSP05:func.2">Dp1</a> (<font color="Maroon" title="c11">m</font>,<font color="Maroon" title="c13">D</font>,<font color="Maroon" title="c21">i</font>)</span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:22_2_1"><span class="lab"><font color="Green" title="E46">A55</font></span></a>, <a class="txt" href="#E19:22_2_1"><span class="lab"><font color="Green" title="E47">B56</font></span></a>, <a class="txt" href="#E4:22_2"><span class="lab"><font color="Green" title="E33">A12</font></span></a></span>;<br/>
<a NAME="E4:22_2_1_2_1_1_4"/><span class="kw">then </span>
<font color="Maroon" title="c22">si</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="absvalue.html#K1" title="ABSVALUE:func.1">sgn</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="dualsp05.html#K3" title="DUALSP05:func.3">Dp2</a> (<font color="Maroon" title="c12">rho</font>,<font color="Maroon" title="c13">D</font>,<span class="p4">(<span class="default"><font color="Maroon" title="c21">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>)</span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p3">(<span class="default"><a href="dualsp05.html#K3" title="DUALSP05:func.3">Dp2</a> (<font color="Maroon" title="c12">rho</font>,<font color="Maroon" title="c13">D</font>,<font color="Maroon" title="c21">i</font>)</span>)</span></span>)</span></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c33">H</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="dualsp05.html#T5" target="_self" onmouseover="rs('dualsp05/T5')" onmouseout="rh()">Lm1</a></span>;<br/>
<span class="kw">then </span><font color="Maroon" title="c22">si</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c18">t</font> = 
<span class="p1">(<span class="default"><a href="absvalue.html#K1" title="ABSVALUE:func.1">sgn</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="dualsp05.html#K3" title="DUALSP05:func.3">Dp2</a> (<font color="Maroon" title="c12">rho</font>,<font color="Maroon" title="c13">D</font>,<span class="p4">(<span class="default"><font color="Maroon" title="c21">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>)</span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p3">(<span class="default"><a href="dualsp05.html#K3" title="DUALSP05:func.3">Dp2</a> (<font color="Maroon" title="c12">rho</font>,<font color="Maroon" title="c13">D</font>,<font color="Maroon" title="c21">i</font>)</span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c34">h</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c18">t</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="c0sp1.html#T30" onmouseover="rs('c0sp1/T30')" onmouseout="rh()">C0SP1:30</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="absvalue.html#K1" title="ABSVALUE:func.1">sgn</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="dualsp05.html#K3" title="DUALSP05:func.3">Dp2</a> (<font color="Maroon" title="c12">rho</font>,<font color="Maroon" title="c13">D</font>,<span class="p4">(<span class="default"><font color="Maroon" title="c21">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>)</span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p3">(<span class="default"><a href="dualsp05.html#K3" title="DUALSP05:func.3">Dp2</a> (<font color="Maroon" title="c12">rho</font>,<font color="Maroon" title="c13">D</font>,<font color="Maroon" title="c21">i</font>)</span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c32">f1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c18">t</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c31">f0</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c18">t</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:22"><span class="lab"><font color="Green" title="E23">A2</font></span></a>, <a class="txt" href="#E7:22_2_1_2_1_1"><span class="lab"><font color="Green" title="E58">A69</font></span></a>, <a class="txt" href="#E8:22_2_1_2_1_1"><span class="lab"><font color="Green" title="E59">A70</font></span></a>, <a class="ref" href="c0sp1.html#T34" onmouseover="rs('c0sp1/T34')" onmouseout="rh()">C0SP1:34</a></span>
;<br/>
<span class="kw">hence </span><a NAME="E6:22_2_1_2_1_1_4"/>
<font color="Maroon" title="c22">si</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c18">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="absvalue.html#K1" title="ABSVALUE:func.1">sgn</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="dualsp05.html#K3" title="DUALSP05:func.3">Dp2</a> (<font color="Maroon" title="c12">rho</font>,<font color="Maroon" title="c13">D</font>,<span class="p4">(<span class="default"><font color="Maroon" title="c21">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>)</span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p3">(<span class="default"><a href="dualsp05.html#K3" title="DUALSP05:func.3">Dp2</a> (<font color="Maroon" title="c12">rho</font>,<font color="Maroon" title="c13">D</font>,<font color="Maroon" title="c21">i</font>)</span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c32">f1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c18">t</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c31">f0</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c18">t</font></span>)</span></span>)</span>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><a NAME="E17:22_2_1_2_1_1"/><span class="lab"><font color="Green" title="E64">A77</font></span>: 
<font color="Maroon" title="c18">t</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c9">a</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c13">D</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">i</font></span>)</span></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:22_2_1_2_1_1"><span class="lab"><font color="Green" title="E54">A63</font></span></a>, <a class="txt" href="#E6:22_2_1_2_1_1"><span class="lab"><font color="Green" title="E57">A66</font></span></a>, <a class="ref" href="integra1.html#T4" onmouseover="rs('integra1/T4')" onmouseout="rh()">INTEGRA1:4</a>, <a class="txt" href="#E10:22_2_1"><span class="lab"><font color="Green" title="E42">A51</font></span></a></span>;<br/><span class="kw">then </span><span class="lab"><font color="Green" title="E65">A79</font></span>: <font color="Maroon" title="c32">f1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c18">t</font> = 
<span class="p1">(<span class="default"><span class="p2"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c9">a</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c13">D</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">i</font></span>)</span></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> 1</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">t</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_4.html#T16" onmouseover="rs('funct_4/T16')" onmouseout="rh()">FUNCT_4:16</a>, <a class="txt" href="#E1:22_2_1_2"><span class="lab"><font color="Green" title="E52">B0</font></span></a>, <a class="ref" href="xxreal_1.html#T90" onmouseover="rs('xxreal_1/T90')" onmouseout="rh()">XXREAL_1:90</a></span>
<br/>.= 
1
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:22_2_1_2_1_1"><span class="lab"><font color="Green" title="E64">A77</font></span></a>, <a class="ref" href="funcop_1.html#T7" onmouseover="rs('funcop_1/T7')" onmouseout="rh()">FUNCOP_1:7</a></span>
;<br/><a NAME="E19:22_2_1_2_1_1"/>
<font color="Maroon" title="c31">f0</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c18">t</font> <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="txt" href="#E11:22"><span class="lab"><font color="Green" title="E28">A7</font></span></a>, <a class="ref" href="funcop_1.html#T7" onmouseover="rs('funcop_1/T7')" onmouseout="rh()">FUNCOP_1:7</a></span>;<br/><span class="kw">hence </span><a NAME="E20:22_2_1_2_1_1"/>
<font color="Maroon" title="c19">z</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="absvalue.html#K1" title="ABSVALUE:func.1">sgn</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="dualsp05.html#K3" title="DUALSP05:func.3">Dp2</a> (<font color="Maroon" title="c12">rho</font>,<font color="Maroon" title="c13">D</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c21">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>)</span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p2">(<span class="default"><a href="dualsp05.html#K3" title="DUALSP05:func.3">Dp2</a> (<font color="Maroon" title="c12">rho</font>,<font color="Maroon" title="c13">D</font>,<font color="Maroon" title="c21">i</font>)</span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:22_2_1"><span class="lab"><font color="Green" title="E46">A55</font></span></a>, <a class="txt" href="#E16:22_2_1_2_1_1"><span class="lab"><font color="Green" title="E63">A76</font></span></a>, <a class="txt" href="#E18:22_2_1_2_1_1"><span class="lab"><font color="Green" title="E65">A79</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><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="22_2_1_2_1_2"><span class="kw">suppose </span></a><a NAME="E1:22_2_1_2_1_2"/><span class="lab"><font color="Green" title="E53">A80</font></span>: 
<font color="Maroon" title="c21">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 1
 ; <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="c19">z</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="absvalue.html#K1" title="ABSVALUE:func.1">sgn</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="dualsp05.html#K3" title="DUALSP05:func.3">Dp2</a> (<font color="Maroon" title="c12">rho</font>,<font color="Maroon" title="c13">D</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c21">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>)</span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p2">(<span class="default"><a href="dualsp05.html#K3" title="DUALSP05:func.3">Dp2</a> (<font color="Maroon" title="c12">rho</font>,<font color="Maroon" title="c13">D</font>,<font color="Maroon" title="c21">i</font>)</span>)</span></span>)</span></span><br/><div class="add"><a NAME="E2:22_2_1_2_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E54">A82</font></span>: 
<font color="Maroon" title="c13">D</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c21">i</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c9">a</font>,<font color="Maroon" title="c10">b</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:22"><span class="lab"><font color="Green" title="E28">A7</font></span></a>, <a class="txt" href="#E9:22_2_1"><span class="lab"><font color="Green" title="E41">A50</font></span></a>, <a class="ref" href="integra1.html#T7" onmouseover="rs('integra1/T7')" onmouseout="rh()">INTEGRA1:7</a></span>;<br/><a NAME="E3:22_2_1_2_1_2"/><span class="lab"><font color="Green" title="E55">A81</font></span>: 
<font color="Maroon" title="c13">D</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">i</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c9">a</font>,<font color="Maroon" title="c10">b</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:22"><span class="lab"><font color="Green" title="E28">A7</font></span></a>, <a class="txt" href="#E9:22_2_1"><span class="lab"><font color="Green" title="E41">A50</font></span></a>, <a class="ref" href="integra1.html#T6" onmouseover="rs('integra1/T6')" onmouseout="rh()">INTEGRA1:6</a></span>;<br/><a NAME="E4:22_2_1_2_1_2"/><span class="lab"><font color="Green" title="E56">A83</font></span>: 
(  <a href="seq_4.html#K3" title="SEQ_4:func.3">lower_bound</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c13">D</font>,<font color="Maroon" title="c21">i</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">D</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c21">i</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span> &amp;  <a href="seq_4.html#K2" title="SEQ_4:func.2">upper_bound</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c13">D</font>,<font color="Maroon" title="c21">i</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">D</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">i</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:22_2_1"><span class="lab"><font color="Green" title="E41">A50</font></span></a>, <a class="txt" href="#E1:22_2_1_2_1_2"><span class="lab"><font color="Green" title="E53">A80</font></span></a>, <a class="ref" href="integra1.html#D4" onmouseover="rs('integra1/D4')" onmouseout="rh()">INTEGRA1:def 4</a></span>;<br/><a NAME="E5:22_2_1_2_1_2"/>
<font color="Maroon" title="c21">i</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c13">D</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:22_2_1"><span class="lab"><font color="Green" title="E41">A50</font></span></a>, <a class="txt" href="#E1:22_2_1_2_1_2"><span class="lab"><font color="Green" title="E53">A80</font></span></a>, <a class="ref" href="integra1.html#T7" onmouseover="rs('integra1/T7')" onmouseout="rh()">INTEGRA1:7</a></span>;<br/><a NAME="E6:22_2_1_2_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E57">A86</font></span>: 
<font color="Maroon" title="c13">D</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c21">i</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c13">D</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">i</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:22_2_1"><span class="lab"><font color="Green" title="E41">A50</font></span></a>, <a class="ref" href="xreal_1.html#T146" onmouseover="rs('xreal_1/T146')" onmouseout="rh()">XREAL_1:146</a>, <a class="ref" href="valued_0.html#D13" onmouseover="rs('valued_0/D13')" onmouseout="rh()">VALUED_0:def 13</a></span>;<br/><a NAME="E7:22_2_1_2_1_2"/>
( <font color="Maroon" title="c13">D</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c21">i</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">a</font> or <font color="Maroon" title="c13">D</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c21">i</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"><a href="rcomp_1.html#K4" title="RCOMP_1:func.4">].</a><span class="default"><font color="Maroon" title="c9">a</font>,<font color="Maroon" title="c10">b</font></span><a href="rcomp_1.html#K4" title="RCOMP_1:func.4">.]</a></span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:22_2_1_2_1_2"><span class="lab"><font color="Green" title="E53">A80</font></span></a>, <a class="txt" href="#E11:22"><span class="lab"><font color="Green" title="E28">A7</font></span></a>, <a class="txt" href="#E9:22_2_1"><span class="lab"><font color="Green" title="E41">A50</font></span></a>, <a class="ref" href="integra1.html#T7" onmouseover="rs('integra1/T7')" onmouseout="rh()">INTEGRA1:7</a>, <a class="ref" href="xxreal_1.html#T6" onmouseover="rs('xxreal_1/T6')" onmouseout="rh()">XXREAL_1:6</a></span>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="22_2_1_2_1_2_1"><span class="kw">per </span></a><a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:22_2_1_2_1_2_1"/><span class="kw">then </span>
( <font color="Maroon" title="c9">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">D</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c21">i</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span> or <font color="Maroon" title="c9">a</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c13">D</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c21">i</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span> )
 </span><span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_1.html#T2" onmouseover="rs('xxreal_1/T2')" onmouseout="rh()">XXREAL_1:2</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="22_2_1_2_1_2_1_1"><span class="kw">suppose </span></a><a NAME="E1:22_2_1_2_1_2_1_1"/><span class="lab"><font color="Green" title="E58">A88</font></span>: 
<font color="Maroon" title="c9">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">D</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c21">i</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</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="c19">z</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="absvalue.html#K1" title="ABSVALUE:func.1">sgn</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="dualsp05.html#K3" title="DUALSP05:func.3">Dp2</a> (<font color="Maroon" title="c12">rho</font>,<font color="Maroon" title="c13">D</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c21">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>)</span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p2">(<span class="default"><a href="dualsp05.html#K3" title="DUALSP05:func.3">Dp2</a> (<font color="Maroon" title="c12">rho</font>,<font color="Maroon" title="c13">D</font>,<font color="Maroon" title="c21">i</font>)</span>)</span></span>)</span></span><br/><div class="add"><a NAME="E2:22_2_1_2_1_2_1_1"/>
1 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c21">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T8" onmouseover="rs('xreal_1/T8')" onmouseout="rh()">XREAL_1:8</a>, <a class="txt" href="#E20:22_2_1"><span class="lab"><font color="Green" title="E48">A57</font></span></a></span>;<br/><span class="kw">then </span><span class="lab"><font color="Green" title="E59">A90</font></span>:  <a href="dualsp05.html#K2" title="DUALSP05:func.2">Dp1</a> (<font color="Maroon" title="c11">m</font>,<font color="Maroon" title="c13">D</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c21">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>) = 
<font color="Maroon" title="c11">m</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">D</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c21">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E25:22_2_1"><span class="lab"><font color="Green" title="E51">A59</font></span></a>, <a class="ref" href="dualsp05.html#D2" target="_self" onmouseover="rs('dualsp05/D2')" onmouseout="rh()">defDp1</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c9">a</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c13">D</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">i</font></span>)</span></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> 1</span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p1">(<span class="default"><span class="p2"><a href="rcomp_1.html#K4" title="RCOMP_1:func.4">].</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c13">D</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">i</font></span>)</span>,<font color="Maroon" title="c10">b</font></span><a href="rcomp_1.html#K4" title="RCOMP_1:func.4">.]</a></span> <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:22"><span class="lab"><font color="Green" title="E29">A8</font></span></a>, <a class="txt" href="#E3:22_2_1_2_1_2"><span class="lab"><font color="Green" title="E55">A81</font></span></a>, <a class="txt" href="#E1:22_2_1_2_1_2_1_1"><span class="lab"><font color="Green" title="E58">A88</font></span></a>, <a class="txt" href="#E6:22_2_1_2_1_2"><span class="lab"><font color="Green" title="E57">A86</font></span></a></span>
;<br/><span class="lab"><font color="Green" title="E60">A91</font></span>:  <a href="dualsp05.html#K2" title="DUALSP05:func.2">Dp1</a> (<font color="Maroon" title="c11">m</font>,<font color="Maroon" title="c13">D</font>,<font color="Maroon" title="c21">i</font>) = 
<font color="Maroon" title="c11">m</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">D</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c21">i</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E22:22_2_1"><span class="lab"><font color="Green" title="E49">A58</font></span></a>, <a class="txt" href="#E1:22_2_1_2_1_2"><span class="lab"><font color="Green" title="E53">A80</font></span></a>, <a class="ref" href="dualsp05.html#D2" target="_self" onmouseover="rs('dualsp05/D2')" onmouseout="rh()">defDp1</a></span>
<br/>.= 
<span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c9">a</font>,<font color="Maroon" title="c10">b</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:22"><span class="lab"><font color="Green" title="E29">A8</font></span></a>, <a class="txt" href="#E2:22_2_1_2_1_2"><span class="lab"><font color="Green" title="E54">A82</font></span></a>, <a class="txt" href="#E1:22_2_1_2_1_2_1_1"><span class="lab"><font color="Green" title="E58">A88</font></span></a></span>
;<br/><a NAME="E5:22_2_1_2_1_2_1_1"/><span class="lab"><font color="Green" title="E61">A93</font></span>: 
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c9">a</font>,<font color="Maroon" title="c10">b</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">A</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:22"><span class="lab"><font color="Green" title="E28">A7</font></span></a>, <a class="ref" href="funcop_1.html#T13" onmouseover="rs('funcop_1/T13')" onmouseout="rh()">FUNCOP_1:13</a></span>;<br/><a NAME="E6:22_2_1_2_1_2_1_1"/><span class="lab"><font color="Green" title="E62">A94</font></span>: 
( <font color="Maroon" title="c9">a</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c13">D</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">i</font> &amp; <font color="Maroon" title="c13">D</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c10">b</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:22_2_1_2_1_2"><span class="lab"><font color="Green" title="E55">A81</font></span></a>, <a class="ref" href="xxreal_1.html#T1" onmouseover="rs('xxreal_1/T1')" onmouseout="rh()">XXREAL_1:1</a></span>;<br/><span class="lab"><font color="Green" title="E63">A96</font></span>:  <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c9">a</font>,<span class="p4">(<span class="default"><font color="Maroon" title="c13">D</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">i</font></span>)</span></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> 1</span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p2">(<span class="default"><span class="p3"><a href="rcomp_1.html#K4" title="RCOMP_1:func.4">].</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c13">D</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">i</font></span>)</span>,<font color="Maroon" title="c10">b</font></span><a href="rcomp_1.html#K4" title="RCOMP_1:func.4">.]</a></span> <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p2">(<span class="default"><span class="p3"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c9">a</font>,<span class="p4">(<span class="default"><font color="Maroon" title="c13">D</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">i</font></span>)</span></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> 1</span>)</span></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p2">(<span class="default"><span class="p3"><a href="rcomp_1.html#K4" title="RCOMP_1:func.4">].</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c13">D</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">i</font></span>)</span>,<font color="Maroon" title="c10">b</font></span><a href="rcomp_1.html#K4" title="RCOMP_1:func.4">.]</a></span> <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_4.html#D1" onmouseover="rs('funct_4/D1')" onmouseout="rh()">FUNCT_4:def 1</a></span>
<br/>.= 
<font color="Maroon" title="c1">A</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:22"><span class="lab"><font color="Green" title="E28">A7</font></span></a>, <a class="txt" href="#E1:22_2_1_2"><span class="lab"><font color="Green" title="E52">B0</font></span></a>, <a class="txt" href="#E6:22_2_1_2_1_2_1_1"><span class="lab"><font color="Green" title="E62">A94</font></span></a>, <a class="ref" href="xxreal_1.html#T167" onmouseover="rs('xxreal_1/T167')" onmouseout="rh()">XXREAL_1:167</a></span>
;<br/><a NAME="E8:22_2_1_2_1_2_1_1"/>
 <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <span class="p1">(<span class="default"><span class="p2"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c9">a</font>,<font color="Maroon" title="c10">b</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> 
 ;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c31">f0</font> = <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c9">a</font>,<font color="Maroon" title="c10">b</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> as   <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> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:22_2_1_2_1_2_1_1"><span class="lab"><font color="Green" title="E61">A93</font></span></a>, <a class="ref" href="funct_2.html#T2" onmouseover="rs('funct_2/T2')" onmouseout="rh()">FUNCT_2:2</a></span>;<br/><a NAME="E10:22_2_1_2_1_2_1_1"/>
 <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c9">a</font>,<span class="p4">(<span class="default"><font color="Maroon" title="c13">D</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">i</font></span>)</span></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> 1</span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p2">(<span class="default"><span class="p3"><a href="rcomp_1.html#K4" title="RCOMP_1:func.4">].</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c13">D</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">i</font></span>)</span>,<font color="Maroon" title="c10">b</font></span><a href="rcomp_1.html#K4" title="RCOMP_1:func.4">.]</a></span> <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span>)</span></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> 
 ;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c32">f1</font> = <span class="p1">(<span class="default"><span class="p2"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c9">a</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c13">D</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">i</font></span>)</span></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> 1</span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p1">(<span class="default"><span class="p2"><a href="rcomp_1.html#K4" title="RCOMP_1:func.4">].</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c13">D</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">i</font></span>)</span>,<font color="Maroon" title="c10">b</font></span><a href="rcomp_1.html#K4" title="RCOMP_1:func.4">.]</a></span> <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span>)</span> as   <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> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:22_2_1_2_1_2_1_1"><span class="lab"><font color="Green" title="E63">A96</font></span></a>, <a class="ref" href="funct_2.html#T2" onmouseover="rs('funct_2/T2')" onmouseout="rh()">FUNCT_2:2</a></span>;<br/><a NAME="E12:22_2_1_2_1_2_1_1"/><span class="lab"><font color="Green" title="E64">A98</font></span>: 
<font color="Maroon" title="c22">si</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c18">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="absvalue.html#K1" title="ABSVALUE:func.1">sgn</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="dualsp05.html#K3" title="DUALSP05:func.3">Dp2</a> (<font color="Maroon" title="c12">rho</font>,<font color="Maroon" title="c13">D</font>,<span class="p4">(<span class="default"><font color="Maroon" title="c21">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>)</span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p3">(<span class="default"><a href="dualsp05.html#K3" title="DUALSP05:func.3">Dp2</a> (<font color="Maroon" title="c12">rho</font>,<font color="Maroon" title="c13">D</font>,<font color="Maroon" title="c21">i</font>)</span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c32">f1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c18">t</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c31">f0</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c18">t</font></span>)</span></span>)</span>
 <div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="22_2_1_2_1_2_1_1_4"><span class="kw">proof </span></a><div class="add">

<span class="kw">reconsider </span><font color="Maroon" title="c33">H</font> = <span class="p1">(<span class="default"><a href="dualsp05.html#K2" title="DUALSP05:func.2">Dp1</a> (<font color="Maroon" title="c11">m</font>,<font color="Maroon" title="c13">D</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c21">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>)</span>)</span> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <span class="p1">(<span class="default"><a href="dualsp05.html#K2" title="DUALSP05:func.2">Dp1</a> (<font color="Maroon" title="c11">m</font>,<font color="Maroon" title="c13">D</font>,<font color="Maroon" title="c21">i</font>)</span>)</span> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="c0sp1.html#K11" title="C0SP1:func.11">R_Normed_Algebra_of_BoundedFunctions</a> <font color="Maroon" title="c1">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="c34">h</font> = <font color="Maroon" title="c33">H</font> as   <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> <span class="kw">by</span> <span class="lab"><a class="ref" href="dualsp05.html#T11" target="_self" onmouseover="rs('dualsp05/T11')" onmouseout="rh()">LM88</a></span>;<br/>
<a NAME="E3:22_2_1_2_1_2_1_1_4"/>
<font color="Maroon" title="c22">si</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="absvalue.html#K1" title="ABSVALUE:func.1">sgn</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="dualsp05.html#K3" title="DUALSP05:func.3">Dp2</a> (<font color="Maroon" title="c12">rho</font>,<font color="Maroon" title="c13">D</font>,<span class="p4">(<span class="default"><font color="Maroon" title="c21">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>)</span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p3">(<span class="default"><a href="dualsp05.html#K3" title="DUALSP05:func.3">Dp2</a> (<font color="Maroon" title="c12">rho</font>,<font color="Maroon" title="c13">D</font>,<font color="Maroon" title="c21">i</font>)</span>)</span></span>)</span></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="dualsp05.html#K2" title="DUALSP05:func.2">Dp1</a> (<font color="Maroon" title="c11">m</font>,<font color="Maroon" title="c13">D</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c21">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>)</span>)</span> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <span class="p2">(<span class="default"><a href="dualsp05.html#K2" title="DUALSP05:func.2">Dp1</a> (<font color="Maroon" title="c11">m</font>,<font color="Maroon" title="c13">D</font>,<font color="Maroon" title="c21">i</font>)</span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:22_2_1"><span class="lab"><font color="Green" title="E46">A55</font></span></a>, <a class="txt" href="#E19:22_2_1"><span class="lab"><font color="Green" title="E47">B56</font></span></a>, <a class="txt" href="#E4:22_2"><span class="lab"><font color="Green" title="E33">A12</font></span></a></span>;<br/>
<a NAME="E4:22_2_1_2_1_2_1_1_4"/><span class="kw">then </span>
<font color="Maroon" title="c22">si</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="absvalue.html#K1" title="ABSVALUE:func.1">sgn</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="dualsp05.html#K3" title="DUALSP05:func.3">Dp2</a> (<font color="Maroon" title="c12">rho</font>,<font color="Maroon" title="c13">D</font>,<span class="p4">(<span class="default"><font color="Maroon" title="c21">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>)</span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p3">(<span class="default"><a href="dualsp05.html#K3" title="DUALSP05:func.3">Dp2</a> (<font color="Maroon" title="c12">rho</font>,<font color="Maroon" title="c13">D</font>,<font color="Maroon" title="c21">i</font>)</span>)</span></span>)</span></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c33">H</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="dualsp05.html#T5" target="_self" onmouseover="rs('dualsp05/T5')" onmouseout="rh()">Lm1</a></span>;<br/>
<span class="kw">then </span><font color="Maroon" title="c22">si</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c18">t</font> = 
<span class="p1">(<span class="default"><a href="absvalue.html#K1" title="ABSVALUE:func.1">sgn</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="dualsp05.html#K3" title="DUALSP05:func.3">Dp2</a> (<font color="Maroon" title="c12">rho</font>,<font color="Maroon" title="c13">D</font>,<span class="p4">(<span class="default"><font color="Maroon" title="c21">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>)</span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p3">(<span class="default"><a href="dualsp05.html#K3" title="DUALSP05:func.3">Dp2</a> (<font color="Maroon" title="c12">rho</font>,<font color="Maroon" title="c13">D</font>,<font color="Maroon" title="c21">i</font>)</span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c34">h</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c18">t</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="c0sp1.html#T30" onmouseover="rs('c0sp1/T30')" onmouseout="rh()">C0SP1:30</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="absvalue.html#K1" title="ABSVALUE:func.1">sgn</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="dualsp05.html#K3" title="DUALSP05:func.3">Dp2</a> (<font color="Maroon" title="c12">rho</font>,<font color="Maroon" title="c13">D</font>,<span class="p4">(<span class="default"><font color="Maroon" title="c21">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>)</span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p3">(<span class="default"><a href="dualsp05.html#K3" title="DUALSP05:func.3">Dp2</a> (<font color="Maroon" title="c12">rho</font>,<font color="Maroon" title="c13">D</font>,<font color="Maroon" title="c21">i</font>)</span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c32">f1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c18">t</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c31">f0</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c18">t</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:22"><span class="lab"><font color="Green" title="E23">A2</font></span></a>, <a class="txt" href="#E3:22_2_1_2_1_2_1_1"><span class="lab"><font color="Green" title="E59">A90</font></span></a>, <a class="txt" href="#E4:22_2_1_2_1_2_1_1"><span class="lab"><font color="Green" title="E60">A91</font></span></a>, <a class="ref" href="c0sp1.html#T34" onmouseover="rs('c0sp1/T34')" onmouseout="rh()">C0SP1:34</a></span>
;<br/>
<span class="kw">hence </span><a NAME="E6:22_2_1_2_1_2_1_1_4"/>
<font color="Maroon" title="c22">si</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c18">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="absvalue.html#K1" title="ABSVALUE:func.1">sgn</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="dualsp05.html#K3" title="DUALSP05:func.3">Dp2</a> (<font color="Maroon" title="c12">rho</font>,<font color="Maroon" title="c13">D</font>,<span class="p4">(<span class="default"><font color="Maroon" title="c21">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>)</span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p3">(<span class="default"><a href="dualsp05.html#K3" title="DUALSP05:func.3">Dp2</a> (<font color="Maroon" title="c12">rho</font>,<font color="Maroon" title="c13">D</font>,<font color="Maroon" title="c21">i</font>)</span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c32">f1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c18">t</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c31">f0</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c18">t</font></span>)</span></span>)</span>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><a NAME="E13:22_2_1_2_1_2_1_1"/><span class="lab"><font color="Green" title="E65">A99</font></span>: 
<font color="Maroon" title="c18">t</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c13">D</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c21">i</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c13">D</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">i</font></span>)</span></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:22_2_1_2_1_2"><span class="lab"><font color="Green" title="E56">A83</font></span></a>, <a class="ref" href="integra1.html#T4" onmouseover="rs('integra1/T4')" onmouseout="rh()">INTEGRA1:4</a>, <a class="txt" href="#E10:22_2_1"><span class="lab"><font color="Green" title="E42">A51</font></span></a></span>;<br/><a NAME="E14:22_2_1_2_1_2_1_1"/>
( <font color="Maroon" title="c9">a</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c13">D</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c21">i</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span> &amp; <font color="Maroon" title="c13">D</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c21">i</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c10">b</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:22_2_1_2_1_2"><span class="lab"><font color="Green" title="E54">A82</font></span></a>, <a class="ref" href="xxreal_1.html#T1" onmouseover="rs('xxreal_1/T1')" onmouseout="rh()">XXREAL_1:1</a></span>;<br/><a NAME="E15:22_2_1_2_1_2_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E66">B100</font></span>: 
<span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c13">D</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c21">i</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c13">D</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">i</font></span>)</span></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c9">a</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c13">D</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">i</font></span>)</span></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_1.html#T34" onmouseover="rs('xxreal_1/T34')" onmouseout="rh()">XXREAL_1:34</a></span>;<br/><span class="kw">then </span><span class="lab"><font color="Green" title="E67">A102</font></span>: <font color="Maroon" title="c32">f1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c18">t</font> = 
<span class="p1">(<span class="default"><span class="p2"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c9">a</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c13">D</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">i</font></span>)</span></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> 1</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">t</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:22_2_1_2_1_2_1_1"><span class="lab"><font color="Green" title="E65">A99</font></span></a>, <a class="ref" href="funct_4.html#T16" onmouseover="rs('funct_4/T16')" onmouseout="rh()">FUNCT_4:16</a>, <a class="txt" href="#E1:22_2_1_2"><span class="lab"><font color="Green" title="E52">B0</font></span></a>, <a class="ref" href="xxreal_1.html#T90" onmouseover="rs('xxreal_1/T90')" onmouseout="rh()">XXREAL_1:90</a></span>
<br/>.= 
1
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:22_2_1_2_1_2_1_1"><span class="lab"><font color="Green" title="E66">B100</font></span></a>, <a class="txt" href="#E13:22_2_1_2_1_2_1_1"><span class="lab"><font color="Green" title="E65">A99</font></span></a>, <a class="ref" href="funcop_1.html#T7" onmouseover="rs('funcop_1/T7')" onmouseout="rh()">FUNCOP_1:7</a></span>
;<br/><a NAME="E17:22_2_1_2_1_2_1_1"/>
<font color="Maroon" title="c31">f0</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c18">t</font> <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="txt" href="#E11:22"><span class="lab"><font color="Green" title="E28">A7</font></span></a>, <a class="ref" href="funcop_1.html#T7" onmouseover="rs('funcop_1/T7')" onmouseout="rh()">FUNCOP_1:7</a></span>;<br/><span class="kw">hence </span><a NAME="E18:22_2_1_2_1_2_1_1"/>
<font color="Maroon" title="c19">z</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="absvalue.html#K1" title="ABSVALUE:func.1">sgn</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="dualsp05.html#K3" title="DUALSP05:func.3">Dp2</a> (<font color="Maroon" title="c12">rho</font>,<font color="Maroon" title="c13">D</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c21">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>)</span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p2">(<span class="default"><a href="dualsp05.html#K3" title="DUALSP05:func.3">Dp2</a> (<font color="Maroon" title="c12">rho</font>,<font color="Maroon" title="c13">D</font>,<font color="Maroon" title="c21">i</font>)</span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:22_2_1"><span class="lab"><font color="Green" title="E46">A55</font></span></a>, <a class="txt" href="#E12:22_2_1_2_1_2_1_1"><span class="lab"><font color="Green" title="E64">A98</font></span></a>, <a class="txt" href="#E16:22_2_1_2_1_2_1_1"><span class="lab"><font color="Green" title="E67">A102</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><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="22_2_1_2_1_2_1_2"><span class="kw">suppose </span></a><a NAME="E1:22_2_1_2_1_2_1_2"/><span class="lab"><font color="Green" title="E58">A103</font></span>: 
<font color="Maroon" title="c9">a</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c13">D</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c21">i</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</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="c19">z</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="absvalue.html#K1" title="ABSVALUE:func.1">sgn</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="dualsp05.html#K3" title="DUALSP05:func.3">Dp2</a> (<font color="Maroon" title="c12">rho</font>,<font color="Maroon" title="c13">D</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c21">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>)</span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p2">(<span class="default"><a href="dualsp05.html#K3" title="DUALSP05:func.3">Dp2</a> (<font color="Maroon" title="c12">rho</font>,<font color="Maroon" title="c13">D</font>,<font color="Maroon" title="c21">i</font>)</span>)</span></span>)</span></span><br/><div class="add"><a NAME="E2:22_2_1_2_1_2_1_2"/>
1 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c21">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T8" onmouseover="rs('xreal_1/T8')" onmouseout="rh()">XREAL_1:8</a>, <a class="txt" href="#E20:22_2_1"><span class="lab"><font color="Green" title="E48">A57</font></span></a></span>;<br/><span class="kw">then </span><span class="lab"><font color="Green" title="E59">A105</font></span>:  <a href="dualsp05.html#K2" title="DUALSP05:func.2">Dp1</a> (<font color="Maroon" title="c11">m</font>,<font color="Maroon" title="c13">D</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c21">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>) = 
<font color="Maroon" title="c11">m</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">D</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c21">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E25:22_2_1"><span class="lab"><font color="Green" title="E51">A59</font></span></a>, <a class="ref" href="dualsp05.html#D2" target="_self" onmouseover="rs('dualsp05/D2')" onmouseout="rh()">defDp1</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c9">a</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c13">D</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">i</font></span>)</span></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> 1</span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p1">(<span class="default"><span class="p2"><a href="rcomp_1.html#K4" title="RCOMP_1:func.4">].</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c13">D</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">i</font></span>)</span>,<font color="Maroon" title="c10">b</font></span><a href="rcomp_1.html#K4" title="RCOMP_1:func.4">.]</a></span> <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:22"><span class="lab"><font color="Green" title="E29">A8</font></span></a>, <a class="txt" href="#E3:22_2_1_2_1_2"><span class="lab"><font color="Green" title="E55">A81</font></span></a>, <a class="txt" href="#E1:22_2_1_2_1_2_1_2"><span class="lab"><font color="Green" title="E58">A103</font></span></a>, <a class="txt" href="#E6:22_2_1_2_1_2"><span class="lab"><font color="Green" title="E57">A86</font></span></a></span>
;<br/><span class="lab"><font color="Green" title="E60">A106</font></span>:  <a href="dualsp05.html#K2" title="DUALSP05:func.2">Dp1</a> (<font color="Maroon" title="c11">m</font>,<font color="Maroon" title="c13">D</font>,<font color="Maroon" title="c21">i</font>) = 
<font color="Maroon" title="c11">m</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">D</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c21">i</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E22:22_2_1"><span class="lab"><font color="Green" title="E49">A58</font></span></a>, <a class="txt" href="#E1:22_2_1_2_1_2"><span class="lab"><font color="Green" title="E53">A80</font></span></a>, <a class="ref" href="dualsp05.html#D2" target="_self" onmouseover="rs('dualsp05/D2')" onmouseout="rh()">defDp1</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c9">a</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c13">D</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c21">i</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span></span>)</span></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> 1</span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p1">(<span class="default"><span class="p2"><a href="rcomp_1.html#K4" title="RCOMP_1:func.4">].</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c13">D</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c21">i</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c10">b</font></span><a href="rcomp_1.html#K4" title="RCOMP_1:func.4">.]</a></span> <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:22"><span class="lab"><font color="Green" title="E29">A8</font></span></a>, <a class="txt" href="#E2:22_2_1_2_1_2"><span class="lab"><font color="Green" title="E54">A82</font></span></a>, <a class="txt" href="#E1:22_2_1_2_1_2_1_2"><span class="lab"><font color="Green" title="E58">A103</font></span></a></span>
;<br/><a NAME="E5:22_2_1_2_1_2_1_2"/><span class="lab"><font color="Green" title="E61">A108</font></span>: 
( <font color="Maroon" title="c9">a</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c13">D</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">i</font> &amp; <font color="Maroon" title="c13">D</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c10">b</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:22_2_1_2_1_2"><span class="lab"><font color="Green" title="E55">A81</font></span></a>, <a class="ref" href="xxreal_1.html#T1" onmouseover="rs('xxreal_1/T1')" onmouseout="rh()">XXREAL_1:1</a></span>;<br/><a NAME="E6:22_2_1_2_1_2_1_2"/><span class="lab"><font color="Green" title="E62">A109</font></span>: 
( <font color="Maroon" title="c9">a</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c13">D</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c21">i</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span> &amp; <font color="Maroon" title="c13">D</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c21">i</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c10">b</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:22_2_1_2_1_2"><span class="lab"><font color="Green" title="E54">A82</font></span></a>, <a class="ref" href="xxreal_1.html#T1" onmouseover="rs('xxreal_1/T1')" onmouseout="rh()">XXREAL_1:1</a></span>;<br/><span class="lab"><font color="Green" title="E63">A110</font></span>:  <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c9">a</font>,<span class="p4">(<span class="default"><font color="Maroon" title="c13">D</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">i</font></span>)</span></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> 1</span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p2">(<span class="default"><span class="p3"><a href="rcomp_1.html#K4" title="RCOMP_1:func.4">].</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c13">D</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">i</font></span>)</span>,<font color="Maroon" title="c10">b</font></span><a href="rcomp_1.html#K4" title="RCOMP_1:func.4">.]</a></span> <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p2">(<span class="default"><span class="p3"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c9">a</font>,<span class="p4">(<span class="default"><font color="Maroon" title="c13">D</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">i</font></span>)</span></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> 1</span>)</span></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p2">(<span class="default"><span class="p3"><a href="rcomp_1.html#K4" title="RCOMP_1:func.4">].</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c13">D</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">i</font></span>)</span>,<font color="Maroon" title="c10">b</font></span><a href="rcomp_1.html#K4" title="RCOMP_1:func.4">.]</a></span> <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_4.html#D1" onmouseover="rs('funct_4/D1')" onmouseout="rh()">FUNCT_4:def 1</a></span>
<br/>.= 
<font color="Maroon" title="c1">A</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:22"><span class="lab"><font color="Green" title="E28">A7</font></span></a>, <a class="txt" href="#E1:22_2_1_2"><span class="lab"><font color="Green" title="E52">B0</font></span></a>, <a class="txt" href="#E5:22_2_1_2_1_2_1_2"><span class="lab"><font color="Green" title="E61">A108</font></span></a>, <a class="ref" href="xxreal_1.html#T167" onmouseover="rs('xxreal_1/T167')" onmouseout="rh()">XXREAL_1:167</a></span>
;<br/><span class="lab"><font color="Green" title="E64">A111</font></span>:  <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c9">a</font>,<span class="p4">(<span class="default"><font color="Maroon" title="c13">D</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p5">(<span class="default"><font color="Maroon" title="c21">i</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span></span>)</span></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> 1</span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p2">(<span class="default"><span class="p3"><a href="rcomp_1.html#K4" title="RCOMP_1:func.4">].</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c13">D</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p5">(<span class="default"><font color="Maroon" title="c21">i</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c10">b</font></span><a href="rcomp_1.html#K4" title="RCOMP_1:func.4">.]</a></span> <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p2">(<span class="default"><span class="p3"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c9">a</font>,<span class="p4">(<span class="default"><font color="Maroon" title="c13">D</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p5">(<span class="default"><font color="Maroon" title="c21">i</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span></span>)</span></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> 1</span>)</span></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p2">(<span class="default"><span class="p3"><a href="rcomp_1.html#K4" title="RCOMP_1:func.4">].</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c13">D</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p5">(<span class="default"><font color="Maroon" title="c21">i</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c10">b</font></span><a href="rcomp_1.html#K4" title="RCOMP_1:func.4">.]</a></span> <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_4.html#D1" onmouseover="rs('funct_4/D1')" onmouseout="rh()">FUNCT_4:def 1</a></span>
<br/>.= 
<font color="Maroon" title="c1">A</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:22"><span class="lab"><font color="Green" title="E28">A7</font></span></a>, <a class="txt" href="#E1:22_2_1_2"><span class="lab"><font color="Green" title="E52">B0</font></span></a>, <a class="txt" href="#E6:22_2_1_2_1_2_1_2"><span class="lab"><font color="Green" title="E62">A109</font></span></a>, <a class="ref" href="xxreal_1.html#T167" onmouseover="rs('xxreal_1/T167')" onmouseout="rh()">XXREAL_1:167</a></span>
;<br/><a NAME="E9:22_2_1_2_1_2_1_2"/>
 <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c9">a</font>,<span class="p4">(<span class="default"><font color="Maroon" title="c13">D</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">i</font></span>)</span></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> 1</span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p2">(<span class="default"><span class="p3"><a href="rcomp_1.html#K4" title="RCOMP_1:func.4">].</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c13">D</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">i</font></span>)</span>,<font color="Maroon" title="c10">b</font></span><a href="rcomp_1.html#K4" title="RCOMP_1:func.4">.]</a></span> <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span>)</span></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> 
 ;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c31">f1</font> = <span class="p1">(<span class="default"><span class="p2"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c9">a</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c13">D</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">i</font></span>)</span></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> 1</span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p1">(<span class="default"><span class="p2"><a href="rcomp_1.html#K4" title="RCOMP_1:func.4">].</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c13">D</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">i</font></span>)</span>,<font color="Maroon" title="c10">b</font></span><a href="rcomp_1.html#K4" title="RCOMP_1:func.4">.]</a></span> <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span>)</span> as   <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> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:22_2_1_2_1_2_1_2"><span class="lab"><font color="Green" title="E63">A110</font></span></a>, <a class="ref" href="funct_2.html#T2" onmouseover="rs('funct_2/T2')" onmouseout="rh()">FUNCT_2:2</a></span>;<br/><a NAME="E11:22_2_1_2_1_2_1_2"/>
 <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c9">a</font>,<span class="p4">(<span class="default"><font color="Maroon" title="c13">D</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p5">(<span class="default"><font color="Maroon" title="c21">i</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span></span>)</span></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> 1</span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p2">(<span class="default"><span class="p3"><a href="rcomp_1.html#K4" title="RCOMP_1:func.4">].</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c13">D</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p5">(<span class="default"><font color="Maroon" title="c21">i</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c10">b</font></span><a href="rcomp_1.html#K4" title="RCOMP_1:func.4">.]</a></span> <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span>)</span></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> 
 ;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c32">f2</font> = <span class="p1">(<span class="default"><span class="p2"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c9">a</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c13">D</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c21">i</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span></span>)</span></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> 1</span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p1">(<span class="default"><span class="p2"><a href="rcomp_1.html#K4" title="RCOMP_1:func.4">].</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c13">D</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c21">i</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c10">b</font></span><a href="rcomp_1.html#K4" title="RCOMP_1:func.4">.]</a></span> <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span>)</span> as   <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> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:22_2_1_2_1_2_1_2"><span class="lab"><font color="Green" title="E64">A111</font></span></a>, <a class="ref" href="funct_2.html#T2" onmouseover="rs('funct_2/T2')" onmouseout="rh()">FUNCT_2:2</a></span>;<br/><a NAME="E13:22_2_1_2_1_2_1_2"/><span class="lab"><font color="Green" title="E65">A112</font></span>: 
( <font color="Maroon" title="c9">a</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c18">t</font> &amp; <font color="Maroon" title="c18">t</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c10">b</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_1.html#T1" onmouseover="rs('xxreal_1/T1')" onmouseout="rh()">XXREAL_1:1</a>, <a class="txt" href="#E11:22"><span class="lab"><font color="Green" title="E28">A7</font></span></a></span>;<br/><a NAME="E14:22_2_1_2_1_2_1_2"/><span class="lab"><font color="Green" title="E66">A113</font></span>: 
<font color="Maroon" title="c22">si</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c18">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="absvalue.html#K1" title="ABSVALUE:func.1">sgn</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="dualsp05.html#K3" title="DUALSP05:func.3">Dp2</a> (<font color="Maroon" title="c12">rho</font>,<font color="Maroon" title="c13">D</font>,<span class="p4">(<span class="default"><font color="Maroon" title="c21">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>)</span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p3">(<span class="default"><a href="dualsp05.html#K3" title="DUALSP05:func.3">Dp2</a> (<font color="Maroon" title="c12">rho</font>,<font color="Maroon" title="c13">D</font>,<font color="Maroon" title="c21">i</font>)</span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c31">f1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c18">t</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c32">f2</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c18">t</font></span>)</span></span>)</span>
 <div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="22_2_1_2_1_2_1_2_5"><span class="kw">proof </span></a><div class="add">

<span class="kw">reconsider </span><font color="Maroon" title="c33">H</font> = <span class="p1">(<span class="default"><a href="dualsp05.html#K2" title="DUALSP05:func.2">Dp1</a> (<font color="Maroon" title="c11">m</font>,<font color="Maroon" title="c13">D</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c21">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>)</span>)</span> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <span class="p1">(<span class="default"><a href="dualsp05.html#K2" title="DUALSP05:func.2">Dp1</a> (<font color="Maroon" title="c11">m</font>,<font color="Maroon" title="c13">D</font>,<font color="Maroon" title="c21">i</font>)</span>)</span> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="c0sp1.html#K11" title="C0SP1:func.11">R_Normed_Algebra_of_BoundedFunctions</a> <font color="Maroon" title="c1">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="c34">h</font> = <font color="Maroon" title="c33">H</font> as   <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> <span class="kw">by</span> <span class="lab"><a class="ref" href="dualsp05.html#T11" target="_self" onmouseover="rs('dualsp05/T11')" onmouseout="rh()">LM88</a></span>;<br/>
<a NAME="E3:22_2_1_2_1_2_1_2_5"/>
<font color="Maroon" title="c22">si</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="absvalue.html#K1" title="ABSVALUE:func.1">sgn</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="dualsp05.html#K3" title="DUALSP05:func.3">Dp2</a> (<font color="Maroon" title="c12">rho</font>,<font color="Maroon" title="c13">D</font>,<span class="p4">(<span class="default"><font color="Maroon" title="c21">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>)</span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p3">(<span class="default"><a href="dualsp05.html#K3" title="DUALSP05:func.3">Dp2</a> (<font color="Maroon" title="c12">rho</font>,<font color="Maroon" title="c13">D</font>,<font color="Maroon" title="c21">i</font>)</span>)</span></span>)</span></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="dualsp05.html#K2" title="DUALSP05:func.2">Dp1</a> (<font color="Maroon" title="c11">m</font>,<font color="Maroon" title="c13">D</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c21">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>)</span>)</span> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <span class="p2">(<span class="default"><a href="dualsp05.html#K2" title="DUALSP05:func.2">Dp1</a> (<font color="Maroon" title="c11">m</font>,<font color="Maroon" title="c13">D</font>,<font color="Maroon" title="c21">i</font>)</span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:22_2_1"><span class="lab"><font color="Green" title="E46">A55</font></span></a>, <a class="txt" href="#E19:22_2_1"><span class="lab"><font color="Green" title="E47">B56</font></span></a>, <a class="txt" href="#E4:22_2"><span class="lab"><font color="Green" title="E33">A12</font></span></a></span>;<br/>
<a NAME="E4:22_2_1_2_1_2_1_2_5"/><span class="kw">then </span>
<font color="Maroon" title="c22">si</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="absvalue.html#K1" title="ABSVALUE:func.1">sgn</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="dualsp05.html#K3" title="DUALSP05:func.3">Dp2</a> (<font color="Maroon" title="c12">rho</font>,<font color="Maroon" title="c13">D</font>,<span class="p4">(<span class="default"><font color="Maroon" title="c21">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>)</span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p3">(<span class="default"><a href="dualsp05.html#K3" title="DUALSP05:func.3">Dp2</a> (<font color="Maroon" title="c12">rho</font>,<font color="Maroon" title="c13">D</font>,<font color="Maroon" title="c21">i</font>)</span>)</span></span>)</span></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c33">H</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="dualsp05.html#T5" target="_self" onmouseover="rs('dualsp05/T5')" onmouseout="rh()">Lm1</a></span>;<br/>
<span class="kw">then </span><font color="Maroon" title="c22">si</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c18">t</font> = 
<span class="p1">(<span class="default"><a href="absvalue.html#K1" title="ABSVALUE:func.1">sgn</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="dualsp05.html#K3" title="DUALSP05:func.3">Dp2</a> (<font color="Maroon" title="c12">rho</font>,<font color="Maroon" title="c13">D</font>,<span class="p4">(<span class="default"><font color="Maroon" title="c21">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>)</span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p3">(<span class="default"><a href="dualsp05.html#K3" title="DUALSP05:func.3">Dp2</a> (<font color="Maroon" title="c12">rho</font>,<font color="Maroon" title="c13">D</font>,<font color="Maroon" title="c21">i</font>)</span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c34">h</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c18">t</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="c0sp1.html#T30" onmouseover="rs('c0sp1/T30')" onmouseout="rh()">C0SP1:30</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="absvalue.html#K1" title="ABSVALUE:func.1">sgn</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="dualsp05.html#K3" title="DUALSP05:func.3">Dp2</a> (<font color="Maroon" title="c12">rho</font>,<font color="Maroon" title="c13">D</font>,<span class="p4">(<span class="default"><font color="Maroon" title="c21">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>)</span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p3">(<span class="default"><a href="dualsp05.html#K3" title="DUALSP05:func.3">Dp2</a> (<font color="Maroon" title="c12">rho</font>,<font color="Maroon" title="c13">D</font>,<font color="Maroon" title="c21">i</font>)</span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c31">f1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c18">t</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c32">f2</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c18">t</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:22"><span class="lab"><font color="Green" title="E23">A2</font></span></a>, <a class="txt" href="#E3:22_2_1_2_1_2_1_2"><span class="lab"><font color="Green" title="E59">A105</font></span></a>, <a class="txt" href="#E4:22_2_1_2_1_2_1_2"><span class="lab"><font color="Green" title="E60">A106</font></span></a>, <a class="ref" href="c0sp1.html#T34" onmouseover="rs('c0sp1/T34')" onmouseout="rh()">C0SP1:34</a></span>
;<br/>
<span class="kw">hence </span><a NAME="E6:22_2_1_2_1_2_1_2_5"/>
<font color="Maroon" title="c22">si</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c18">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="absvalue.html#K1" title="ABSVALUE:func.1">sgn</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="dualsp05.html#K3" title="DUALSP05:func.3">Dp2</a> (<font color="Maroon" title="c12">rho</font>,<font color="Maroon" title="c13">D</font>,<span class="p4">(<span class="default"><font color="Maroon" title="c21">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>)</span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p3">(<span class="default"><a href="dualsp05.html#K3" title="DUALSP05:func.3">Dp2</a> (<font color="Maroon" title="c12">rho</font>,<font color="Maroon" title="c13">D</font>,<font color="Maroon" title="c21">i</font>)</span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c31">f1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c18">t</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c32">f2</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c18">t</font></span>)</span></span>)</span>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><a NAME="E15:22_2_1_2_1_2_1_2"/><span class="lab"><font color="Green" title="E67">A114</font></span>: 
<font color="Maroon" title="c18">t</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c13">D</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c21">i</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c13">D</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">i</font></span>)</span></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:22_2_1_2_1_2"><span class="lab"><font color="Green" title="E56">A83</font></span></a>, <a class="ref" href="integra1.html#T4" onmouseover="rs('integra1/T4')" onmouseout="rh()">INTEGRA1:4</a>, <a class="txt" href="#E10:22_2_1"><span class="lab"><font color="Green" title="E42">A51</font></span></a></span>;<br/><a NAME="E16:22_2_1_2_1_2_1_2"/><span class="lab"><font color="Green" title="E68">B115</font></span>: 
<span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c13">D</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c21">i</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c13">D</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">i</font></span>)</span></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c9">a</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c13">D</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">i</font></span>)</span></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:22_2_1_2_1_2_1_2"><span class="lab"><font color="Green" title="E62">A109</font></span></a>, <a class="ref" href="xxreal_1.html#T34" onmouseover="rs('xxreal_1/T34')" onmouseout="rh()">XXREAL_1:34</a></span>;<br/><span class="kw">then </span><span class="lab"><font color="Green" title="E69">A117</font></span>: <font color="Maroon" title="c31">f1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c18">t</font> = 
<span class="p1">(<span class="default"><span class="p2"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c9">a</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c13">D</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">i</font></span>)</span></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> 1</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">t</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:22_2_1_2_1_2_1_2"><span class="lab"><font color="Green" title="E67">A114</font></span></a>, <a class="ref" href="funct_4.html#T16" onmouseover="rs('funct_4/T16')" onmouseout="rh()">FUNCT_4:16</a>, <a class="txt" href="#E1:22_2_1_2"><span class="lab"><font color="Green" title="E52">B0</font></span></a>, <a class="ref" href="xxreal_1.html#T90" onmouseover="rs('xxreal_1/T90')" onmouseout="rh()">XXREAL_1:90</a></span>
<br/>.= 
1
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:22_2_1_2_1_2_1_2"><span class="lab"><font color="Green" title="E68">B115</font></span></a>, <a class="txt" href="#E15:22_2_1_2_1_2_1_2"><span class="lab"><font color="Green" title="E67">A114</font></span></a>, <a class="ref" href="funcop_1.html#T7" onmouseover="rs('funcop_1/T7')" onmouseout="rh()">FUNCOP_1:7</a></span>
;<br/><a NAME="E18:22_2_1_2_1_2_1_2"/>
( <font color="Maroon" title="c13">D</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c21">i</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c18">t</font> &amp; <font color="Maroon" title="c18">t</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c13">D</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">i</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:22_2_1"><span class="lab"><font color="Green" title="E41">A50</font></span></a>, <a class="txt" href="#E1:22_2_1_2_1_2"><span class="lab"><font color="Green" title="E53">A80</font></span></a>, <a class="ref" href="integra1.html#D4" onmouseover="rs('integra1/D4')" onmouseout="rh()">INTEGRA1:def 4</a>, <a class="txt" href="#E11:22_2_1"><span class="lab"><font color="Green" title="E43">A52</font></span></a></span>;<br/><a NAME="E19:22_2_1_2_1_2_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E70">A118</font></span>: 
<font color="Maroon" title="c18">t</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"><a href="rcomp_1.html#K4" title="RCOMP_1:func.4">].</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c13">D</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c21">i</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c10">b</font></span><a href="rcomp_1.html#K4" title="RCOMP_1:func.4">.]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:22_2_1_2_1_2_1_2"><span class="lab"><font color="Green" title="E65">A112</font></span></a></span>;<br/><span class="kw">then </span><font color="Maroon" title="c32">f2</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c18">t</font> = 
<span class="p1">(<span class="default"><span class="p2"><a href="rcomp_1.html#K4" title="RCOMP_1:func.4">].</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c13">D</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c21">i</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c10">b</font></span><a href="rcomp_1.html#K4" title="RCOMP_1:func.4">.]</a></span> <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">t</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:22_2_1_2"><span class="lab"><font color="Green" title="E52">B0</font></span></a>, <a class="ref" href="funct_4.html#T13" onmouseover="rs('funct_4/T13')" onmouseout="rh()">FUNCT_4:13</a></span>
<br/>.= 
 <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:22_2_1_2_1_2_1_2"><span class="lab"><font color="Green" title="E70">A118</font></span></a>, <a class="ref" href="funcop_1.html#T7" onmouseover="rs('funcop_1/T7')" onmouseout="rh()">FUNCOP_1:7</a></span>
;<br/><span class="kw">hence </span><a NAME="E21:22_2_1_2_1_2_1_2"/>
<font color="Maroon" title="c19">z</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="absvalue.html#K1" title="ABSVALUE:func.1">sgn</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="dualsp05.html#K3" title="DUALSP05:func.3">Dp2</a> (<font color="Maroon" title="c12">rho</font>,<font color="Maroon" title="c13">D</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c21">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>)</span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p2">(<span class="default"><a href="dualsp05.html#K3" title="DUALSP05:func.3">Dp2</a> (<font color="Maroon" title="c12">rho</font>,<font color="Maroon" title="c13">D</font>,<font color="Maroon" title="c21">i</font>)</span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:22_2_1"><span class="lab"><font color="Green" title="E46">A55</font></span></a>, <a class="txt" href="#E14:22_2_1_2_1_2_1_2"><span class="lab"><font color="Green" title="E66">A113</font></span></a>, <a class="txt" href="#E17:22_2_1_2_1_2_1_2"><span class="lab"><font color="Green" title="E69">A117</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></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

</div><span class="kw">end;</span></div>
<a NAME="E27:22_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E52">A119</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="c19">z</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">i</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> 1
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="dualsp05.html#T1" target="_self" onmouseover="rs('dualsp05/T1')" onmouseout="rh()">LM91</a></span>;<br/>
<a NAME="E28:22_2_1"/>
 for <font color="Olive" title="b1">k</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Olive" title="b1">k</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> <font color="Maroon" title="c19">z</font> &amp; <font color="Olive" title="b1">k</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c21">i</font> holds <br/><font color="Maroon" title="c19">z</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> 
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="22_2_1_3"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c24">k</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="c24">k</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> <font color="Maroon" title="c19">z</font> &amp; <font color="Maroon" title="c24">k</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c21">i</font> implies <font color="Maroon" title="c19">z</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c24">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:22_2_1_3"/><span class="lab"><font color="Green" title="E53">A120</font></span>: 
<font color="Maroon" title="c24">k</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> <font color="Maroon" title="c19">z</font>
 <span class="kw">and </span><br/><a NAME="E2:22_2_1_3"/><span class="lab"><font color="Green" title="E54">A121</font></span>: 
<font color="Maroon" title="c24">k</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c21">i</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="c19">z</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c24">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> </span><br/>

<span class="kw">consider </span><font color="Maroon" title="c25">sk</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><span class="kw"> such that </span><br/><a NAME="E4:22_2_1_3"/><span class="lab"><font color="Green" title="E55">A124</font></span>: 
( <font color="Maroon" title="c25">sk</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">s</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c24">k</font> &amp; <font color="Maroon" title="c19">z</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c24">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c25">sk</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c18">t</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:22_2_1"><span class="lab"><font color="Green" title="E38">A18</font></span></a>, <a class="txt" href="#E1:22_2_1_3"><span class="lab"><font color="Green" title="E53">A120</font></span></a></span>;<br/>
<a NAME="E5:22_2_1_3"/><span class="lab"><font color="Green" title="E56">B125</font></span>: 
<font color="Maroon" title="c24">k</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> <font color="Maroon" title="c15">s</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>, <a class="txt" href="#E3:22_2"><span class="lab"><font color="Green" title="E32">A11</font></span></a>, <a class="txt" href="#E3:22_2_1"><span class="lab"><font color="Green" title="E38">A18</font></span></a>, <a class="txt" href="#E1:22_2_1_3"><span class="lab"><font color="Green" title="E53">A120</font></span></a></span>;<br/>
<a NAME="E6:22_2_1_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E57">A125</font></span>: 
<font color="Maroon" title="c15">s</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c24">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="absvalue.html#K1" title="ABSVALUE:func.1">sgn</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="dualsp05.html#K3" title="DUALSP05:func.3">Dp2</a> (<font color="Maroon" title="c12">rho</font>,<font color="Maroon" title="c13">D</font>,<span class="p4">(<span class="default"><font color="Maroon" title="c24">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>)</span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p3">(<span class="default"><a href="dualsp05.html#K3" title="DUALSP05:func.3">Dp2</a> (<font color="Maroon" title="c12">rho</font>,<font color="Maroon" title="c13">D</font>,<font color="Maroon" title="c24">k</font>)</span>)</span></span>)</span></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="dualsp05.html#K2" title="DUALSP05:func.2">Dp1</a> (<font color="Maroon" title="c11">m</font>,<font color="Maroon" title="c13">D</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c24">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>)</span>)</span> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <span class="p2">(<span class="default"><a href="dualsp05.html#K2" title="DUALSP05:func.2">Dp1</a> (<font color="Maroon" title="c11">m</font>,<font color="Maroon" title="c13">D</font>,<font color="Maroon" title="c24">k</font>)</span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:22_2"><span class="lab"><font color="Green" title="E33">A12</font></span></a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c26">r</font> =  <a href="absvalue.html#K1" title="ABSVALUE:func.1">sgn</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="dualsp05.html#K3" title="DUALSP05:func.3">Dp2</a> (<font color="Maroon" title="c12">rho</font>,<font color="Maroon" title="c13">D</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c24">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>)</span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p2">(<span class="default"><a href="dualsp05.html#K3" title="DUALSP05:func.3">Dp2</a> (<font color="Maroon" title="c12">rho</font>,<font color="Maroon" title="c13">D</font>,<font color="Maroon" title="c24">k</font>)</span>)</span></span>)</span>;<br/>
<a NAME="E7:22_2_1_3"/><span class="lab"><font color="Green" title="E58">A126</font></span>: 
<font color="Maroon" title="c24">k</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> <font color="Maroon" title="c13">D</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:22_2_1"><span class="lab"><font color="Green" title="E38">A18</font></span></a>, <a class="txt" href="#E1:22_2_1_3"><span class="lab"><font color="Green" title="E53">A120</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="E8:22_2_1_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E59">A127</font></span>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c24">k</font> &amp; <font color="Maroon" title="c24">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c13">D</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_3.html#T25" onmouseover="rs('finseq_3/T25')" onmouseout="rh()">FINSEQ_3:25</a></span>;<br/>
<a NAME="E9:22_2_1_3"/><span class="kw">then </span>
<font color="Maroon" title="c24">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c13">D</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T7" onmouseover="rs('xreal_1/T7')" onmouseout="rh()">XREAL_1:7</a></span>;<br/>
<a NAME="E10:22_2_1_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E60">A128</font></span>: 
<font color="Maroon" title="c24">k</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"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c13">D</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:22_2_1_3"><span class="lab"><font color="Green" title="E59">A127</font></span></a></span>;<br/>
<a NAME="E11:22_2_1_3"/><span class="lab"><font color="Green" title="E61">a</font></span>: 
1 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c24">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T7" onmouseover="rs('xreal_1/T7')" onmouseout="rh()">XREAL_1:7</a></span>;<br/>
<a NAME="E12:22_2_1_3"/>
<font color="Maroon" title="c24">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c13">D</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:22_2_1_3"><span class="lab"><font color="Green" title="E59">A127</font></span></a>, <a class="ref" href="xreal_1.html#T7" onmouseover="rs('xreal_1/T7')" onmouseout="rh()">XREAL_1:7</a></span>;<br/>
<a NAME="E13:22_2_1_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E62">A129</font></span>: 
<font color="Maroon" title="c24">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <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"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c13">D</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:22_2_1_3"><span class="lab"><font color="Green" title="E61">a</font></span></a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c27">f0</font> = <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c9">a</font>,<font color="Maroon" title="c10">b</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>;<br/>
<span class="kw">set </span><font color="Maroon" title="c28">f1</font> = <span class="p1">(<span class="default"><span class="p2"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c9">a</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c13">D</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c24">k</font></span>)</span></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> 1</span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p1">(<span class="default"><span class="p2"><a href="rcomp_1.html#K4" title="RCOMP_1:func.4">].</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c13">D</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c24">k</font></span>)</span>,<font color="Maroon" title="c10">b</font></span><a href="rcomp_1.html#K4" title="RCOMP_1:func.4">.]</a></span> <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span>)</span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c29">g1</font> = <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c9">a</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c13">D</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c24">k</font></span>)</span></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> 1;<br/>
<span class="kw">set </span><font color="Maroon" title="c30">g2</font> = <span class="p1"><a href="rcomp_1.html#K4" title="RCOMP_1:func.4">].</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c13">D</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c24">k</font></span>)</span>,<font color="Maroon" title="c10">b</font></span><a href="rcomp_1.html#K4" title="RCOMP_1:func.4">.]</a></span> <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>;<br/>
<span class="kw">set </span><font color="Maroon" title="c31">f2</font> = <span class="p1">(<span class="default"><span class="p2"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c9">a</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c13">D</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c24">k</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span></span>)</span></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> 1</span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p1">(<span class="default"><span class="p2"><a href="rcomp_1.html#K4" title="RCOMP_1:func.4">].</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c13">D</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c24">k</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c10">b</font></span><a href="rcomp_1.html#K4" title="RCOMP_1:func.4">.]</a></span> <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span>)</span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c32">h1</font> = <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c9">a</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c13">D</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c24">k</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span></span>)</span></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> 1;<br/>
<span class="kw">set </span><font color="Maroon" title="c33">h2</font> = <span class="p1"><a href="rcomp_1.html#K4" title="RCOMP_1:func.4">].</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c13">D</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c24">k</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c10">b</font></span><a href="rcomp_1.html#K4" title="RCOMP_1:func.4">.]</a></span> <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>;<br/>
<a NAME="E14:22_2_1_3"/><span class="lab"><font color="Green" title="E63">B10</font></span>: 
(  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c9">a</font>,<font color="Maroon" title="c10">b</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c9">a</font>,<font color="Maroon" title="c10">b</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> &amp;  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c9">a</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c13">D</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c24">k</font></span>)</span></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c9">a</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c13">D</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c24">k</font></span>)</span></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> &amp;  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2"><a href="rcomp_1.html#K4" title="RCOMP_1:func.4">].</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c13">D</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c24">k</font></span>)</span>,<font color="Maroon" title="c10">b</font></span><a href="rcomp_1.html#K4" title="RCOMP_1:func.4">.]</a></span> <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="rcomp_1.html#K4" title="RCOMP_1:func.4">].</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c13">D</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c24">k</font></span>)</span>,<font color="Maroon" title="c10">b</font></span><a href="rcomp_1.html#K4" title="RCOMP_1:func.4">.]</a></span> &amp;  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c9">a</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c13">D</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c24">k</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span></span>)</span></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c9">a</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c13">D</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c24">k</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span></span>)</span></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> &amp;  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2"><a href="rcomp_1.html#K4" title="RCOMP_1:func.4">].</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c13">D</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c24">k</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c10">b</font></span><a href="rcomp_1.html#K4" title="RCOMP_1:func.4">.]</a></span> <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="rcomp_1.html#K4" title="RCOMP_1:func.4">].</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c13">D</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c24">k</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c10">b</font></span><a href="rcomp_1.html#K4" title="RCOMP_1:func.4">.]</a></span> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funcop_1.html#T13" onmouseover="rs('funcop_1/T13')" onmouseout="rh()">FUNCOP_1:13</a></span>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="22_2_1_3_1"><span class="kw">per </span></a><a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:22_2_1_3_1"/>
( <font color="Maroon" title="c24">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or <font color="Maroon" title="c24">k</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 1 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="22_2_1_3_1_1"><span class="kw">suppose </span></a><a NAME="E1:22_2_1_3_1_1"/><span class="lab"><font color="Green" title="E64">A130</font></span>: 
<font color="Maroon" title="c24">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 ; <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="c19">z</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c24">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> </span><br/><div class="add"><a NAME="E2:22_2_1_3_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E65">A134</font></span>: 
(  <a href="seq_4.html#K3" title="SEQ_4:func.3">lower_bound</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c13">D</font>,<font color="Maroon" title="c21">i</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">D</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c21">i</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span> &amp;  <a href="seq_4.html#K2" title="SEQ_4:func.2">upper_bound</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c13">D</font>,<font color="Maroon" title="c21">i</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">D</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">i</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:22_2_1"><span class="lab"><font color="Green" title="E41">A50</font></span></a>, <a class="ref" href="integra1.html#D4" onmouseover="rs('integra1/D4')" onmouseout="rh()">INTEGRA1:def 4</a>, <a class="txt" href="#E2:22_2_1_3"><span class="lab"><font color="Green" title="E54">A121</font></span></a></span>;<br/><a NAME="E3:22_2_1_3_1_1"/><span class="lab"><font color="Green" title="E66">A136</font></span>: 
<font color="Maroon" title="c21">i</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c13">D</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:22_2_1"><span class="lab"><font color="Green" title="E41">A50</font></span></a>, <a class="ref" href="integra1.html#T7" onmouseover="rs('integra1/T7')" onmouseout="rh()">INTEGRA1:7</a>, <a class="txt" href="#E1:22_2_1_3_1_1"><span class="lab"><font color="Green" title="E64">A130</font></span></a>, <a class="txt" href="#E2:22_2_1_3"><span class="lab"><font color="Green" title="E54">A121</font></span></a></span>;<br/><a NAME="E4:22_2_1_3_1_1"/><span class="lab"><font color="Green" title="E67">A141</font></span>: 
<font color="Maroon" title="c13">D</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c24">k</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c9">a</font>,<font color="Maroon" title="c10">b</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:22"><span class="lab"><font color="Green" title="E28">A7</font></span></a>, <a class="txt" href="#E7:22_2_1_3"><span class="lab"><font color="Green" title="E58">A126</font></span></a>, <a class="ref" href="integra1.html#T6" onmouseover="rs('integra1/T6')" onmouseout="rh()">INTEGRA1:6</a></span>;<br/><a NAME="E5:22_2_1_3_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E68">A147</font></span>: 
( <font color="Maroon" title="c9">a</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c13">D</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c24">k</font> &amp; <font color="Maroon" title="c13">D</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c24">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c10">b</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_1.html#T1" onmouseover="rs('xxreal_1/T1')" onmouseout="rh()">XXREAL_1:1</a></span>;<br/><a NAME="E6:22_2_1_3_1_1"/><span class="lab"><font color="Green" title="E69">A146</font></span>: 
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c9">a</font>,<font color="Maroon" title="c10">b</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">A</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:22"><span class="lab"><font color="Green" title="E28">A7</font></span></a>, <a class="ref" href="funcop_1.html#T13" onmouseover="rs('funcop_1/T13')" onmouseout="rh()">FUNCOP_1:13</a></span>;<br/><span class="lab"><font color="Green" title="E70">A148</font></span>:  <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c9">a</font>,<span class="p4">(<span class="default"><font color="Maroon" title="c13">D</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c24">k</font></span>)</span></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> 1</span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p2">(<span class="default"><span class="p3"><a href="rcomp_1.html#K4" title="RCOMP_1:func.4">].</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c13">D</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c24">k</font></span>)</span>,<font color="Maroon" title="c10">b</font></span><a href="rcomp_1.html#K4" title="RCOMP_1:func.4">.]</a></span> <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p2">(<span class="default"><span class="p3"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c9">a</font>,<span class="p4">(<span class="default"><font color="Maroon" title="c13">D</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c24">k</font></span>)</span></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> 1</span>)</span></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p2">(<span class="default"><span class="p3"><a href="rcomp_1.html#K4" title="RCOMP_1:func.4">].</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c13">D</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c24">k</font></span>)</span>,<font color="Maroon" title="c10">b</font></span><a href="rcomp_1.html#K4" title="RCOMP_1:func.4">.]</a></span> <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_4.html#D1" onmouseover="rs('funct_4/D1')" onmouseout="rh()">FUNCT_4:def 1</a></span>
<br/>.= 
<font color="Maroon" title="c1">A</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:22"><span class="lab"><font color="Green" title="E28">A7</font></span></a>, <a class="txt" href="#E14:22_2_1_3"><span class="lab"><font color="Green" title="E63">B10</font></span></a>, <a class="txt" href="#E5:22_2_1_3_1_1"><span class="lab"><font color="Green" title="E68">A147</font></span></a>, <a class="ref" href="xxreal_1.html#T167" onmouseover="rs('xxreal_1/T167')" onmouseout="rh()">XXREAL_1:167</a></span>
;<br/><a NAME="E8:22_2_1_3_1_1"/>
 <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <span class="p1">(<span class="default"><span class="p2"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c9">a</font>,<font color="Maroon" title="c10">b</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> 
 ;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c34">f0</font> = <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c9">a</font>,<font color="Maroon" title="c10">b</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> as   <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> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:22_2_1_3_1_1"><span class="lab"><font color="Green" title="E69">A146</font></span></a>, <a class="ref" href="funct_2.html#T2" onmouseover="rs('funct_2/T2')" onmouseout="rh()">FUNCT_2:2</a></span>;<br/><a NAME="E10:22_2_1_3_1_1"/>
 <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c9">a</font>,<span class="p4">(<span class="default"><font color="Maroon" title="c13">D</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c24">k</font></span>)</span></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> 1</span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p2">(<span class="default"><span class="p3"><a href="rcomp_1.html#K4" title="RCOMP_1:func.4">].</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c13">D</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c24">k</font></span>)</span>,<font color="Maroon" title="c10">b</font></span><a href="rcomp_1.html#K4" title="RCOMP_1:func.4">.]</a></span> <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span>)</span></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> 
 ;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c35">f1</font> = <span class="p1">(<span class="default"><span class="p2"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c9">a</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c13">D</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c24">k</font></span>)</span></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> 1</span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p1">(<span class="default"><span class="p2"><a href="rcomp_1.html#K4" title="RCOMP_1:func.4">].</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c13">D</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c24">k</font></span>)</span>,<font color="Maroon" title="c10">b</font></span><a href="rcomp_1.html#K4" title="RCOMP_1:func.4">.]</a></span> <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span>)</span> as   <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> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:22_2_1_3_1_1"><span class="lab"><font color="Green" title="E70">A148</font></span></a>, <a class="ref" href="funct_2.html#T2" onmouseover="rs('funct_2/T2')" onmouseout="rh()">FUNCT_2:2</a></span>;<br/><span class="lab"><font color="Green" title="E71">A153</font></span>:  <a href="dualsp05.html#K2" title="DUALSP05:func.2">Dp1</a> (<font color="Maroon" title="c11">m</font>,<font color="Maroon" title="c13">D</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c24">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>) = 
<font color="Maroon" title="c11">m</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">D</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c24">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:22_2_1_3"><span class="lab"><font color="Green" title="E62">A129</font></span></a>, <a class="ref" href="dualsp05.html#D2" target="_self" onmouseover="rs('dualsp05/D2')" onmouseout="rh()">defDp1</a>, <a class="txt" href="#E1:22_2_1_3_1_1"><span class="lab"><font color="Green" title="E64">A130</font></span></a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c9">a</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c13">D</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c24">k</font></span>)</span></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> 1</span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p1">(<span class="default"><span class="p2"><a href="rcomp_1.html#K4" title="RCOMP_1:func.4">].</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c13">D</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c24">k</font></span>)</span>,<font color="Maroon" title="c10">b</font></span><a href="rcomp_1.html#K4" title="RCOMP_1:func.4">.]</a></span> <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:22"><span class="lab"><font color="Green" title="E29">A8</font></span></a>, <a class="txt" href="#E4:22_2_1_3_1_1"><span class="lab"><font color="Green" title="E67">A141</font></span></a>, <a class="txt" href="#E1:22_2"><span class="lab"><font color="Green" title="E31">A10</font></span></a>, <a class="txt" href="#E1:22_2_1_3_1_1"><span class="lab"><font color="Green" title="E64">A130</font></span></a></span>
;<br/><a NAME="E13:22_2_1_3_1_1"/>
<font color="Maroon" title="c1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><span class="p2">(<span class="default"><a href="seq_4.html#K3" title="SEQ_4:func.3">lower_bound</a> <font color="Maroon" title="c1">A</font></span>)</span>,<span class="p2">(<span class="default"><a href="seq_4.html#K2" title="SEQ_4:func.2">upper_bound</a> <font color="Maroon" title="c1">A</font></span>)</span></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="integra1.html#T4" onmouseover="rs('integra1/T4')" onmouseout="rh()">INTEGRA1:4</a></span>;<br/><a NAME="E14:22_2_1_3_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E72">A139</font></span>: 
 <a href="seq_4.html#K3" title="SEQ_4:func.3">lower_bound</a> <font color="Maroon" title="c1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">a</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:22"><span class="lab"><font color="Green" title="E28">A7</font></span></a>, <a class="ref" href="integra1.html#T5" onmouseover="rs('integra1/T5')" onmouseout="rh()">INTEGRA1:5</a></span>;<br/><a NAME="E15:22_2_1_3_1_1"/><span class="lab"><font color="Green" title="E73">A140</font></span>: 
<font color="Maroon" title="c9">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c9">a</font>,<font color="Maroon" title="c10">b</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:22"><span class="lab"><font color="Green" title="E28">A7</font></span></a></span>;<br/><span class="lab"><font color="Green" title="E74">A144</font></span>:  <a href="dualsp05.html#K2" title="DUALSP05:func.2">Dp1</a> (<font color="Maroon" title="c11">m</font>,<font color="Maroon" title="c13">D</font>,<font color="Maroon" title="c24">k</font>) = 
<font color="Maroon" title="c11">m</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="seq_4.html#K3" title="SEQ_4:func.3">lower_bound</a> <font color="Maroon" title="c1">A</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:22_2_1_3"><span class="lab"><font color="Green" title="E60">A128</font></span></a>, <a class="txt" href="#E1:22_2_1_3_1_1"><span class="lab"><font color="Green" title="E64">A130</font></span></a>, <a class="ref" href="dualsp05.html#D2" target="_self" onmouseover="rs('dualsp05/D2')" onmouseout="rh()">defDp1</a></span>
<br/>.= 
<span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c9">a</font>,<font color="Maroon" title="c10">b</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:22"><span class="lab"><font color="Green" title="E29">A8</font></span></a>, <a class="txt" href="#E14:22_2_1_3_1_1"><span class="lab"><font color="Green" title="E72">A139</font></span></a>, <a class="txt" href="#E15:22_2_1_3_1_1"><span class="lab"><font color="Green" title="E73">A140</font></span></a></span>
;<br/><a NAME="E17:22_2_1_3_1_1"/><span class="lab"><font color="Green" title="E75">A154</font></span>: 
<font color="Maroon" title="c25">sk</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c18">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="absvalue.html#K1" title="ABSVALUE:func.1">sgn</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="dualsp05.html#K3" title="DUALSP05:func.3">Dp2</a> (<font color="Maroon" title="c12">rho</font>,<font color="Maroon" title="c13">D</font>,<span class="p4">(<span class="default"><font color="Maroon" title="c24">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>)</span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p3">(<span class="default"><a href="dualsp05.html#K3" title="DUALSP05:func.3">Dp2</a> (<font color="Maroon" title="c12">rho</font>,<font color="Maroon" title="c13">D</font>,<font color="Maroon" title="c24">k</font>)</span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c35">f1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c18">t</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c34">f0</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c18">t</font></span>)</span></span>)</span>
 <div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="22_2_1_3_1_1_4"><span class="kw">proof </span></a><div class="add">

<span class="kw">reconsider </span><font color="Maroon" title="c36">H</font> = <span class="p1">(<span class="default"><a href="dualsp05.html#K2" title="DUALSP05:func.2">Dp1</a> (<font color="Maroon" title="c11">m</font>,<font color="Maroon" title="c13">D</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c24">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>)</span>)</span> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <span class="p1">(<span class="default"><a href="dualsp05.html#K2" title="DUALSP05:func.2">Dp1</a> (<font color="Maroon" title="c11">m</font>,<font color="Maroon" title="c13">D</font>,<font color="Maroon" title="c24">k</font>)</span>)</span> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="c0sp1.html#K11" title="C0SP1:func.11">R_Normed_Algebra_of_BoundedFunctions</a> <font color="Maroon" title="c1">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="c37">h</font> = <font color="Maroon" title="c36">H</font> as   <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> <span class="kw">by</span> <span class="lab"><a class="ref" href="dualsp05.html#T11" target="_self" onmouseover="rs('dualsp05/T11')" onmouseout="rh()">LM88</a></span>;<br/>
<a NAME="E3:22_2_1_3_1_1_4"/>
<font color="Maroon" title="c25">sk</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="absvalue.html#K1" title="ABSVALUE:func.1">sgn</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="dualsp05.html#K3" title="DUALSP05:func.3">Dp2</a> (<font color="Maroon" title="c12">rho</font>,<font color="Maroon" title="c13">D</font>,<span class="p4">(<span class="default"><font color="Maroon" title="c24">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>)</span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p3">(<span class="default"><a href="dualsp05.html#K3" title="DUALSP05:func.3">Dp2</a> (<font color="Maroon" title="c12">rho</font>,<font color="Maroon" title="c13">D</font>,<font color="Maroon" title="c24">k</font>)</span>)</span></span>)</span></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="dualsp05.html#K2" title="DUALSP05:func.2">Dp1</a> (<font color="Maroon" title="c11">m</font>,<font color="Maroon" title="c13">D</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c24">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>)</span>)</span> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <span class="p2">(<span class="default"><a href="dualsp05.html#K2" title="DUALSP05:func.2">Dp1</a> (<font color="Maroon" title="c11">m</font>,<font color="Maroon" title="c13">D</font>,<font color="Maroon" title="c24">k</font>)</span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:22_2_1_3"><span class="lab"><font color="Green" title="E55">A124</font></span></a>, <a class="txt" href="#E5:22_2_1_3"><span class="lab"><font color="Green" title="E56">B125</font></span></a>, <a class="txt" href="#E4:22_2"><span class="lab"><font color="Green" title="E33">A12</font></span></a></span>;<br/>
<a NAME="E4:22_2_1_3_1_1_4"/><span class="kw">then </span>
<font color="Maroon" title="c25">sk</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="absvalue.html#K1" title="ABSVALUE:func.1">sgn</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="dualsp05.html#K3" title="DUALSP05:func.3">Dp2</a> (<font color="Maroon" title="c12">rho</font>,<font color="Maroon" title="c13">D</font>,<span class="p4">(<span class="default"><font color="Maroon" title="c24">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>)</span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p3">(<span class="default"><a href="dualsp05.html#K3" title="DUALSP05:func.3">Dp2</a> (<font color="Maroon" title="c12">rho</font>,<font color="Maroon" title="c13">D</font>,<font color="Maroon" title="c24">k</font>)</span>)</span></span>)</span></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c36">H</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="dualsp05.html#T5" target="_self" onmouseover="rs('dualsp05/T5')" onmouseout="rh()">Lm1</a></span>;<br/>
<span class="kw">then </span><font color="Maroon" title="c25">sk</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c18">t</font> = 
<span class="p1">(<span class="default"><a href="absvalue.html#K1" title="ABSVALUE:func.1">sgn</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="dualsp05.html#K3" title="DUALSP05:func.3">Dp2</a> (<font color="Maroon" title="c12">rho</font>,<font color="Maroon" title="c13">D</font>,<span class="p4">(<span class="default"><font color="Maroon" title="c24">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>)</span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p3">(<span class="default"><a href="dualsp05.html#K3" title="DUALSP05:func.3">Dp2</a> (<font color="Maroon" title="c12">rho</font>,<font color="Maroon" title="c13">D</font>,<font color="Maroon" title="c24">k</font>)</span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c37">h</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c18">t</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="c0sp1.html#T30" onmouseover="rs('c0sp1/T30')" onmouseout="rh()">C0SP1:30</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="absvalue.html#K1" title="ABSVALUE:func.1">sgn</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="dualsp05.html#K3" title="DUALSP05:func.3">Dp2</a> (<font color="Maroon" title="c12">rho</font>,<font color="Maroon" title="c13">D</font>,<span class="p4">(<span class="default"><font color="Maroon" title="c24">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>)</span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p3">(<span class="default"><a href="dualsp05.html#K3" title="DUALSP05:func.3">Dp2</a> (<font color="Maroon" title="c12">rho</font>,<font color="Maroon" title="c13">D</font>,<font color="Maroon" title="c24">k</font>)</span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c35">f1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c18">t</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c34">f0</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c18">t</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:22"><span class="lab"><font color="Green" title="E23">A2</font></span></a>, <a class="txt" href="#E16:22_2_1_3_1_1"><span class="lab"><font color="Green" title="E74">A144</font></span></a>, <a class="txt" href="#E12:22_2_1_3_1_1"><span class="lab"><font color="Green" title="E71">A153</font></span></a>, <a class="ref" href="c0sp1.html#T34" onmouseover="rs('c0sp1/T34')" onmouseout="rh()">C0SP1:34</a></span>
;<br/>
<span class="kw">hence </span><a NAME="E6:22_2_1_3_1_1_4"/>
<font color="Maroon" title="c25">sk</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c18">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="absvalue.html#K1" title="ABSVALUE:func.1">sgn</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="dualsp05.html#K3" title="DUALSP05:func.3">Dp2</a> (<font color="Maroon" title="c12">rho</font>,<font color="Maroon" title="c13">D</font>,<span class="p4">(<span class="default"><font color="Maroon" title="c24">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>)</span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p3">(<span class="default"><a href="dualsp05.html#K3" title="DUALSP05:func.3">Dp2</a> (<font color="Maroon" title="c12">rho</font>,<font color="Maroon" title="c13">D</font>,<font color="Maroon" title="c24">k</font>)</span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c35">f1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c18">t</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c34">f0</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c18">t</font></span>)</span></span>)</span>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><a NAME="E18:22_2_1_3_1_1"/>
<font color="Maroon" title="c24">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c21">i</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:22_2_1_3_1_1"><span class="lab"><font color="Green" title="E64">A130</font></span></a>, <a class="txt" href="#E2:22_2_1_3"><span class="lab"><font color="Green" title="E54">A121</font></span></a>, <a class="txt" href="#E20:22_2_1"><span class="lab"><font color="Green" title="E48">A57</font></span></a>, <a class="ref" href="xxreal_0.html#T1" onmouseover="rs('xxreal_0/T1')" onmouseout="rh()">XXREAL_0:1</a></span>;<br/><a NAME="E19:22_2_1_3_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c24">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c21">i</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T13" onmouseover="rs('nat_1/T13')" onmouseout="rh()">NAT_1:13</a></span>;<br/><a NAME="E20:22_2_1_3_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E76">A157</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c24">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c21">i</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T13" onmouseover="rs('xreal_1/T13')" onmouseout="rh()">XREAL_1:13</a></span>;<br/><a NAME="E21:22_2_1_3_1_1"/>
<font color="Maroon" title="c13">D</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c24">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c13">D</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c21">i</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span>
 <div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="22_2_1_3_1_1_5"><span class="kw">proof </span></a><div class="add">

<a NAME="E1:22_2_1_3_1_1_5"/>
( <font color="Maroon" title="c24">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">i</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 or <font color="Maroon" title="c24">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c21">i</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:22_2_1_3_1_1"><span class="lab"><font color="Green" title="E76">A157</font></span></a>, <a class="ref" href="xxreal_0.html#T1" onmouseover="rs('xxreal_0/T1')" onmouseout="rh()">XXREAL_0:1</a></span>;<br/>
<span class="kw">hence </span><a NAME="E2:22_2_1_3_1_1_5"/>
<font color="Maroon" title="c13">D</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c24">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c13">D</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c21">i</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:22_2_1_3"><span class="lab"><font color="Green" title="E58">A126</font></span></a>, <a class="txt" href="#E3:22_2_1_3_1_1"><span class="lab"><font color="Green" title="E66">A136</font></span></a>, <a class="ref" href="valued_0.html#D13" onmouseover="rs('valued_0/D13')" onmouseout="rh()">VALUED_0:def 13</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="E22:22_2_1_3_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E77">A158</font></span>: 
<font color="Maroon" title="c13">D</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c24">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c18">t</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:22_2_1"><span class="lab"><font color="Green" title="E43">A52</font></span></a>, <a class="txt" href="#E1:22_2_1_3_1_1"><span class="lab"><font color="Green" title="E64">A130</font></span></a>, <a class="txt" href="#E2:22_2_1_3"><span class="lab"><font color="Green" title="E54">A121</font></span></a>, <a class="ref" href="xxreal_0.html#T2" onmouseover="rs('xxreal_0/T2')" onmouseout="rh()">XXREAL_0:2</a>, <a class="txt" href="#E2:22_2_1_3_1_1"><span class="lab"><font color="Green" title="E65">A134</font></span></a></span>;<br/><a NAME="E23:22_2_1_3_1_1"/>
( <font color="Maroon" title="c9">a</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c18">t</font> &amp; <font color="Maroon" title="c18">t</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c10">b</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_1.html#T1" onmouseover="rs('xxreal_1/T1')" onmouseout="rh()">XXREAL_1:1</a>, <a class="txt" href="#E11:22"><span class="lab"><font color="Green" title="E28">A7</font></span></a></span>;<br/><a NAME="E24:22_2_1_3_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E78">A160</font></span>: 
<font color="Maroon" title="c18">t</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"><a href="rcomp_1.html#K4" title="RCOMP_1:func.4">].</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c13">D</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c24">k</font></span>)</span>,<font color="Maroon" title="c10">b</font></span><a href="rcomp_1.html#K4" title="RCOMP_1:func.4">.]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E22:22_2_1_3_1_1"><span class="lab"><font color="Green" title="E77">A158</font></span></a></span>;<br/><span class="kw">then </span><span class="lab"><font color="Green" title="E79">A162</font></span>: <font color="Maroon" title="c35">f1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c18">t</font> = 
<span class="p1">(<span class="default"><span class="p2"><a href="rcomp_1.html#K4" title="RCOMP_1:func.4">].</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c13">D</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c24">k</font></span>)</span>,<font color="Maroon" title="c10">b</font></span><a href="rcomp_1.html#K4" title="RCOMP_1:func.4">.]</a></span> <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">t</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:22_2_1_3"><span class="lab"><font color="Green" title="E63">B10</font></span></a>, <a class="ref" href="funct_4.html#T13" onmouseover="rs('funct_4/T13')" onmouseout="rh()">FUNCT_4:13</a></span>
<br/>.= 
 <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E24:22_2_1_3_1_1"><span class="lab"><font color="Green" title="E78">A160</font></span></a>, <a class="ref" href="funcop_1.html#T7" onmouseover="rs('funcop_1/T7')" onmouseout="rh()">FUNCOP_1:7</a></span>
;<br/><a NAME="E26:22_2_1_3_1_1"/>
<font color="Maroon" title="c34">f0</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c18">t</font> <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="txt" href="#E11:22"><span class="lab"><font color="Green" title="E28">A7</font></span></a>, <a class="ref" href="funcop_1.html#T7" onmouseover="rs('funcop_1/T7')" onmouseout="rh()">FUNCOP_1:7</a></span>;<br/><span class="kw">hence </span><a NAME="E27:22_2_1_3_1_1"/>
<font color="Maroon" title="c19">z</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c24">k</font> <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="txt" href="#E4:22_2_1_3"><span class="lab"><font color="Green" title="E55">A124</font></span></a>, <a class="txt" href="#E17:22_2_1_3_1_1"><span class="lab"><font color="Green" title="E75">A154</font></span></a>, <a class="txt" href="#E25:22_2_1_3_1_1"><span class="lab"><font color="Green" title="E79">A162</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><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="22_2_1_3_1_2"><span class="kw">suppose </span></a><a NAME="E1:22_2_1_3_1_2"/><span class="lab"><font color="Green" title="E64">A163</font></span>: 
<font color="Maroon" title="c24">k</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 1
 ; <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="c19">z</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c24">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> </span><br/><div class="add"><a NAME="E2:22_2_1_3_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E65">A165</font></span>: 
<font color="Maroon" title="c13">D</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c24">k</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c9">a</font>,<font color="Maroon" title="c10">b</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:22"><span class="lab"><font color="Green" title="E28">A7</font></span></a>, <a class="txt" href="#E7:22_2_1_3"><span class="lab"><font color="Green" title="E58">A126</font></span></a>, <a class="ref" href="integra1.html#T7" onmouseover="rs('integra1/T7')" onmouseout="rh()">INTEGRA1:7</a></span>;<br/><a NAME="E3:22_2_1_3_1_2"/><span class="lab"><font color="Green" title="E66">A164</font></span>: 
<font color="Maroon" title="c13">D</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c24">k</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c9">a</font>,<font color="Maroon" title="c10">b</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:22"><span class="lab"><font color="Green" title="E28">A7</font></span></a>, <a class="txt" href="#E7:22_2_1_3"><span class="lab"><font color="Green" title="E58">A126</font></span></a>, <a class="ref" href="integra1.html#T6" onmouseover="rs('integra1/T6')" onmouseout="rh()">INTEGRA1:6</a></span>;<br/><a NAME="E4:22_2_1_3_1_2"/><span class="lab"><font color="Green" title="E67">A168</font></span>: 
<font color="Maroon" title="c24">k</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c13">D</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:22_2_1_3"><span class="lab"><font color="Green" title="E58">A126</font></span></a>, <a class="txt" href="#E1:22_2_1_3_1_2"><span class="lab"><font color="Green" title="E64">A163</font></span></a>, <a class="ref" href="integra1.html#T7" onmouseover="rs('integra1/T7')" onmouseout="rh()">INTEGRA1:7</a></span>;<br/><a NAME="E5:22_2_1_3_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E68">A169</font></span>: 
<font color="Maroon" title="c13">D</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c24">k</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c13">D</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c24">k</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:22_2_1_3"><span class="lab"><font color="Green" title="E58">A126</font></span></a>, <a class="ref" href="xreal_1.html#T146" onmouseover="rs('xreal_1/T146')" onmouseout="rh()">XREAL_1:146</a>, <a class="ref" href="valued_0.html#D13" onmouseover="rs('valued_0/D13')" onmouseout="rh()">VALUED_0:def 13</a></span>;<br/><a NAME="E6:22_2_1_3_1_2"/>
1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c24">k</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:22_2_1_3"><span class="lab"><font color="Green" title="E59">A127</font></span></a>, <a class="txt" href="#E1:22_2_1_3_1_2"><span class="lab"><font color="Green" title="E64">A163</font></span></a>, <a class="ref" href="xxreal_0.html#T1" onmouseover="rs('xxreal_0/T1')" onmouseout="rh()">XXREAL_0:1</a></span>;<br/><a NAME="E7:22_2_1_3_1_2"/><span class="kw">then </span>
1 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c24">k</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T13" onmouseover="rs('nat_1/T13')" onmouseout="rh()">NAT_1:13</a></span>;<br/><a NAME="E8:22_2_1_3_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E69">A171</font></span>: 
2 <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c24">k</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T13" onmouseover="rs('xreal_1/T13')" onmouseout="rh()">XREAL_1:13</a></span>;<br/><a NAME="E9:22_2_1_3_1_2"/>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c13">D</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:22_2_1_3"><span class="lab"><font color="Green" title="E59">A127</font></span></a>, <a class="ref" href="xxreal_0.html#T2" onmouseover="rs('xxreal_0/T2')" onmouseout="rh()">XXREAL_0:2</a></span>;<br/><a NAME="E10:22_2_1_3_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E70">A172</font></span>: 
1 <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c13">D</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_3.html#T25" onmouseover="rs('finseq_3/T25')" onmouseout="rh()">FINSEQ_3:25</a></span>;<br/><a NAME="E11:22_2_1_3_1_2"/><span class="lab"><font color="Green" title="E71">B173</font></span>: 
<font color="Maroon" title="c13">D</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c13">D</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c24">k</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span>
 <div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="22_2_1_3_1_2_1"><span class="kw">proof </span></a><div class="add">

<a NAME="E1:22_2_1_3_1_2_1"/>
( 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">k</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 or 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c24">k</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:22_2_1_3_1_2"><span class="lab"><font color="Green" title="E69">A171</font></span></a>, <a class="ref" href="xxreal_0.html#T1" onmouseover="rs('xxreal_0/T1')" onmouseout="rh()">XXREAL_0:1</a></span>;<br/>
<span class="kw">hence </span><a NAME="E2:22_2_1_3_1_2_1"/>
<font color="Maroon" title="c13">D</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c13">D</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c24">k</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:22_2_1_3_1_2"><span class="lab"><font color="Green" title="E67">A168</font></span></a>, <a class="txt" href="#E10:22_2_1_3_1_2"><span class="lab"><font color="Green" title="E70">A172</font></span></a>, <a class="ref" href="valued_0.html#D13" onmouseover="rs('valued_0/D13')" onmouseout="rh()">VALUED_0:def 13</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="E12:22_2_1_3_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E72">A173</font></span>: 
<font color="Maroon" title="c9">a</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c13">D</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c24">k</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:22_2"><span class="lab"><font color="Green" title="E31">A10</font></span></a>, <a class="ref" href="xxreal_0.html#T2" onmouseover="rs('xxreal_0/T2')" onmouseout="rh()">XXREAL_0:2</a></span>;<br/><a NAME="E13:22_2_1_3_1_2"/>
1 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c24">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T8" onmouseover="rs('xreal_1/T8')" onmouseout="rh()">XREAL_1:8</a>, <a class="txt" href="#E8:22_2_1_3"><span class="lab"><font color="Green" title="E59">A127</font></span></a></span>;<br/><span class="kw">then </span><span class="lab"><font color="Green" title="E73">A175</font></span>:  <a href="dualsp05.html#K2" title="DUALSP05:func.2">Dp1</a> (<font color="Maroon" title="c11">m</font>,<font color="Maroon" title="c13">D</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c24">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>) = 
<font color="Maroon" title="c11">m</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">D</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c24">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:22_2_1_3"><span class="lab"><font color="Green" title="E62">A129</font></span></a>, <a class="ref" href="dualsp05.html#D2" target="_self" onmouseover="rs('dualsp05/D2')" onmouseout="rh()">defDp1</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c9">a</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c13">D</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c24">k</font></span>)</span></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> 1</span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p1">(<span class="default"><span class="p2"><a href="rcomp_1.html#K4" title="RCOMP_1:func.4">].</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c13">D</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c24">k</font></span>)</span>,<font color="Maroon" title="c10">b</font></span><a href="rcomp_1.html#K4" title="RCOMP_1:func.4">.]</a></span> <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:22"><span class="lab"><font color="Green" title="E29">A8</font></span></a>, <a class="txt" href="#E3:22_2_1_3_1_2"><span class="lab"><font color="Green" title="E66">A164</font></span></a>, <a class="txt" href="#E12:22_2_1_3_1_2"><span class="lab"><font color="Green" title="E72">A173</font></span></a>, <a class="txt" href="#E5:22_2_1_3_1_2"><span class="lab"><font color="Green" title="E68">A169</font></span></a></span>
;<br/><span class="lab"><font color="Green" title="E74">A176</font></span>:  <a href="dualsp05.html#K2" title="DUALSP05:func.2">Dp1</a> (<font color="Maroon" title="c11">m</font>,<font color="Maroon" title="c13">D</font>,<font color="Maroon" title="c24">k</font>) = 
<font color="Maroon" title="c11">m</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">D</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c24">k</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:22_2_1_3"><span class="lab"><font color="Green" title="E60">A128</font></span></a>, <a class="txt" href="#E1:22_2_1_3_1_2"><span class="lab"><font color="Green" title="E64">A163</font></span></a>, <a class="ref" href="dualsp05.html#D2" target="_self" onmouseover="rs('dualsp05/D2')" onmouseout="rh()">defDp1</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c9">a</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c13">D</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c24">k</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span></span>)</span></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> 1</span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p1">(<span class="default"><span class="p2"><a href="rcomp_1.html#K4" title="RCOMP_1:func.4">].</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c13">D</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c24">k</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c10">b</font></span><a href="rcomp_1.html#K4" title="RCOMP_1:func.4">.]</a></span> <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:22"><span class="lab"><font color="Green" title="E29">A8</font></span></a>, <a class="txt" href="#E2:22_2_1_3_1_2"><span class="lab"><font color="Green" title="E65">A165</font></span></a>, <a class="txt" href="#E11:22_2_1_3_1_2"><span class="lab"><font color="Green" title="E71">B173</font></span></a>, <a class="txt" href="#E1:22_2"><span class="lab"><font color="Green" title="E31">A10</font></span></a></span>
;<br/><a NAME="E16:22_2_1_3_1_2"/><span class="lab"><font color="Green" title="E75">A178</font></span>: 
( <font color="Maroon" title="c9">a</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c13">D</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c24">k</font> &amp; <font color="Maroon" title="c13">D</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c24">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c10">b</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:22_2_1_3_1_2"><span class="lab"><font color="Green" title="E66">A164</font></span></a>, <a class="ref" href="xxreal_1.html#T1" onmouseover="rs('xxreal_1/T1')" onmouseout="rh()">XXREAL_1:1</a></span>;<br/><a NAME="E17:22_2_1_3_1_2"/><span class="lab"><font color="Green" title="E76">A179</font></span>: 
( <font color="Maroon" title="c9">a</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c13">D</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c24">k</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span> &amp; <font color="Maroon" title="c13">D</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c24">k</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c10">b</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:22_2_1_3_1_2"><span class="lab"><font color="Green" title="E65">A165</font></span></a>, <a class="ref" href="xxreal_1.html#T1" onmouseover="rs('xxreal_1/T1')" onmouseout="rh()">XXREAL_1:1</a></span>;<br/><span class="lab"><font color="Green" title="E77">A180</font></span>:  <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c9">a</font>,<span class="p4">(<span class="default"><font color="Maroon" title="c13">D</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c24">k</font></span>)</span></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> 1</span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p2">(<span class="default"><span class="p3"><a href="rcomp_1.html#K4" title="RCOMP_1:func.4">].</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c13">D</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c24">k</font></span>)</span>,<font color="Maroon" title="c10">b</font></span><a href="rcomp_1.html#K4" title="RCOMP_1:func.4">.]</a></span> <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p2">(<span class="default"><span class="p3"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c9">a</font>,<span class="p4">(<span class="default"><font color="Maroon" title="c13">D</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c24">k</font></span>)</span></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> 1</span>)</span></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p2">(<span class="default"><span class="p3"><a href="rcomp_1.html#K4" title="RCOMP_1:func.4">].</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c13">D</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c24">k</font></span>)</span>,<font color="Maroon" title="c10">b</font></span><a href="rcomp_1.html#K4" title="RCOMP_1:func.4">.]</a></span> <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_4.html#D1" onmouseover="rs('funct_4/D1')" onmouseout="rh()">FUNCT_4:def 1</a></span>
<br/>.= 
<font color="Maroon" title="c1">A</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:22"><span class="lab"><font color="Green" title="E28">A7</font></span></a>, <a class="txt" href="#E14:22_2_1_3"><span class="lab"><font color="Green" title="E63">B10</font></span></a>, <a class="txt" href="#E16:22_2_1_3_1_2"><span class="lab"><font color="Green" title="E75">A178</font></span></a>, <a class="ref" href="xxreal_1.html#T167" onmouseover="rs('xxreal_1/T167')" onmouseout="rh()">XXREAL_1:167</a></span>
;<br/><span class="lab"><font color="Green" title="E78">A181</font></span>:  <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c9">a</font>,<span class="p4">(<span class="default"><font color="Maroon" title="c13">D</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p5">(<span class="default"><font color="Maroon" title="c24">k</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span></span>)</span></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> 1</span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p2">(<span class="default"><span class="p3"><a href="rcomp_1.html#K4" title="RCOMP_1:func.4">].</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c13">D</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p5">(<span class="default"><font color="Maroon" title="c24">k</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c10">b</font></span><a href="rcomp_1.html#K4" title="RCOMP_1:func.4">.]</a></span> <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p2">(<span class="default"><span class="p3"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c9">a</font>,<span class="p4">(<span class="default"><font color="Maroon" title="c13">D</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p5">(<span class="default"><font color="Maroon" title="c24">k</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span></span>)</span></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> 1</span>)</span></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p2">(<span class="default"><span class="p3"><a href="rcomp_1.html#K4" title="RCOMP_1:func.4">].</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c13">D</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p5">(<span class="default"><font color="Maroon" title="c24">k</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c10">b</font></span><a href="rcomp_1.html#K4" title="RCOMP_1:func.4">.]</a></span> <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_4.html#D1" onmouseover="rs('funct_4/D1')" onmouseout="rh()">FUNCT_4:def 1</a></span>
<br/>.= 
<font color="Maroon" title="c1">A</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:22"><span class="lab"><font color="Green" title="E28">A7</font></span></a>, <a class="txt" href="#E14:22_2_1_3"><span class="lab"><font color="Green" title="E63">B10</font></span></a>, <a class="txt" href="#E17:22_2_1_3_1_2"><span class="lab"><font color="Green" title="E76">A179</font></span></a>, <a class="ref" href="xxreal_1.html#T167" onmouseover="rs('xxreal_1/T167')" onmouseout="rh()">XXREAL_1:167</a></span>
;<br/><a NAME="E20:22_2_1_3_1_2"/>
 <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c9">a</font>,<span class="p4">(<span class="default"><font color="Maroon" title="c13">D</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c24">k</font></span>)</span></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> 1</span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p2">(<span class="default"><span class="p3"><a href="rcomp_1.html#K4" title="RCOMP_1:func.4">].</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c13">D</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c24">k</font></span>)</span>,<font color="Maroon" title="c10">b</font></span><a href="rcomp_1.html#K4" title="RCOMP_1:func.4">.]</a></span> <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span>)</span></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> 
 ;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c34">f1</font> = <span class="p1">(<span class="default"><span class="p2"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c9">a</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c13">D</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c24">k</font></span>)</span></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> 1</span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p1">(<span class="default"><span class="p2"><a href="rcomp_1.html#K4" title="RCOMP_1:func.4">].</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c13">D</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c24">k</font></span>)</span>,<font color="Maroon" title="c10">b</font></span><a href="rcomp_1.html#K4" title="RCOMP_1:func.4">.]</a></span> <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span>)</span> as   <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> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:22_2_1_3_1_2"><span class="lab"><font color="Green" title="E77">A180</font></span></a>, <a class="ref" href="funct_2.html#T2" onmouseover="rs('funct_2/T2')" onmouseout="rh()">FUNCT_2:2</a></span>;<br/><a NAME="E22:22_2_1_3_1_2"/>
 <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c9">a</font>,<span class="p4">(<span class="default"><font color="Maroon" title="c13">D</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p5">(<span class="default"><font color="Maroon" title="c24">k</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span></span>)</span></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> 1</span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p2">(<span class="default"><span class="p3"><a href="rcomp_1.html#K4" title="RCOMP_1:func.4">].</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c13">D</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p5">(<span class="default"><font color="Maroon" title="c24">k</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c10">b</font></span><a href="rcomp_1.html#K4" title="RCOMP_1:func.4">.]</a></span> <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span>)</span></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> 
 ;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c35">f2</font> = <span class="p1">(<span class="default"><span class="p2"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c9">a</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c13">D</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c24">k</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span></span>)</span></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> 1</span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p1">(<span class="default"><span class="p2"><a href="rcomp_1.html#K4" title="RCOMP_1:func.4">].</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c13">D</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c24">k</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c10">b</font></span><a href="rcomp_1.html#K4" title="RCOMP_1:func.4">.]</a></span> <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span>)</span> as   <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> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:22_2_1_3_1_2"><span class="lab"><font color="Green" title="E78">A181</font></span></a>, <a class="ref" href="funct_2.html#T2" onmouseover="rs('funct_2/T2')" onmouseout="rh()">FUNCT_2:2</a></span>;<br/><a NAME="E24:22_2_1_3_1_2"/><span class="lab"><font color="Green" title="E79">A183</font></span>: 
<font color="Maroon" title="c25">sk</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c18">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="absvalue.html#K1" title="ABSVALUE:func.1">sgn</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="dualsp05.html#K3" title="DUALSP05:func.3">Dp2</a> (<font color="Maroon" title="c12">rho</font>,<font color="Maroon" title="c13">D</font>,<span class="p4">(<span class="default"><font color="Maroon" title="c24">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>)</span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p3">(<span class="default"><a href="dualsp05.html#K3" title="DUALSP05:func.3">Dp2</a> (<font color="Maroon" title="c12">rho</font>,<font color="Maroon" title="c13">D</font>,<font color="Maroon" title="c24">k</font>)</span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c34">f1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c18">t</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c35">f2</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c18">t</font></span>)</span></span>)</span>
 <div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="22_2_1_3_1_2_6"><span class="kw">proof </span></a><div class="add">

<span class="kw">reconsider </span><font color="Maroon" title="c36">H</font> = <span class="p1">(<span class="default"><a href="dualsp05.html#K2" title="DUALSP05:func.2">Dp1</a> (<font color="Maroon" title="c11">m</font>,<font color="Maroon" title="c13">D</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c24">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>)</span>)</span> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <span class="p1">(<span class="default"><a href="dualsp05.html#K2" title="DUALSP05:func.2">Dp1</a> (<font color="Maroon" title="c11">m</font>,<font color="Maroon" title="c13">D</font>,<font color="Maroon" title="c24">k</font>)</span>)</span> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="c0sp1.html#K11" title="C0SP1:func.11">R_Normed_Algebra_of_BoundedFunctions</a> <font color="Maroon" title="c1">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="c37">h</font> = <font color="Maroon" title="c36">H</font> as   <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> <span class="kw">by</span> <span class="lab"><a class="ref" href="dualsp05.html#T11" target="_self" onmouseover="rs('dualsp05/T11')" onmouseout="rh()">LM88</a></span>;<br/>
<a NAME="E3:22_2_1_3_1_2_6"/>
<font color="Maroon" title="c25">sk</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="absvalue.html#K1" title="ABSVALUE:func.1">sgn</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="dualsp05.html#K3" title="DUALSP05:func.3">Dp2</a> (<font color="Maroon" title="c12">rho</font>,<font color="Maroon" title="c13">D</font>,<span class="p4">(<span class="default"><font color="Maroon" title="c24">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>)</span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p3">(<span class="default"><a href="dualsp05.html#K3" title="DUALSP05:func.3">Dp2</a> (<font color="Maroon" title="c12">rho</font>,<font color="Maroon" title="c13">D</font>,<font color="Maroon" title="c24">k</font>)</span>)</span></span>)</span></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c36">H</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="dualsp05.html#T5" target="_self" onmouseover="rs('dualsp05/T5')" onmouseout="rh()">Lm1</a>, <a class="txt" href="#E4:22_2_1_3"><span class="lab"><font color="Green" title="E55">A124</font></span></a>, <a class="txt" href="#E6:22_2_1_3"><span class="lab"><font color="Green" title="E57">A125</font></span></a></span>;<br/>
<span class="kw">then </span><font color="Maroon" title="c25">sk</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c18">t</font> = 
<span class="p1">(<span class="default"><a href="absvalue.html#K1" title="ABSVALUE:func.1">sgn</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="dualsp05.html#K3" title="DUALSP05:func.3">Dp2</a> (<font color="Maroon" title="c12">rho</font>,<font color="Maroon" title="c13">D</font>,<span class="p4">(<span class="default"><font color="Maroon" title="c24">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>)</span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p3">(<span class="default"><a href="dualsp05.html#K3" title="DUALSP05:func.3">Dp2</a> (<font color="Maroon" title="c12">rho</font>,<font color="Maroon" title="c13">D</font>,<font color="Maroon" title="c24">k</font>)</span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c37">h</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c18">t</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="c0sp1.html#T30" onmouseover="rs('c0sp1/T30')" onmouseout="rh()">C0SP1:30</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="absvalue.html#K1" title="ABSVALUE:func.1">sgn</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="dualsp05.html#K3" title="DUALSP05:func.3">Dp2</a> (<font color="Maroon" title="c12">rho</font>,<font color="Maroon" title="c13">D</font>,<span class="p4">(<span class="default"><font color="Maroon" title="c24">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>)</span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p3">(<span class="default"><a href="dualsp05.html#K3" title="DUALSP05:func.3">Dp2</a> (<font color="Maroon" title="c12">rho</font>,<font color="Maroon" title="c13">D</font>,<font color="Maroon" title="c24">k</font>)</span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c34">f1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c18">t</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c35">f2</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c18">t</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:22"><span class="lab"><font color="Green" title="E23">A2</font></span></a>, <a class="txt" href="#E14:22_2_1_3_1_2"><span class="lab"><font color="Green" title="E73">A175</font></span></a>, <a class="txt" href="#E15:22_2_1_3_1_2"><span class="lab"><font color="Green" title="E74">A176</font></span></a>, <a class="ref" href="c0sp1.html#T34" onmouseover="rs('c0sp1/T34')" onmouseout="rh()">C0SP1:34</a></span>
;<br/>
<span class="kw">hence </span><a NAME="E5:22_2_1_3_1_2_6"/>
<font color="Maroon" title="c25">sk</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c18">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="absvalue.html#K1" title="ABSVALUE:func.1">sgn</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="dualsp05.html#K3" title="DUALSP05:func.3">Dp2</a> (<font color="Maroon" title="c12">rho</font>,<font color="Maroon" title="c13">D</font>,<span class="p4">(<span class="default"><font color="Maroon" title="c24">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>)</span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p3">(<span class="default"><a href="dualsp05.html#K3" title="DUALSP05:func.3">Dp2</a> (<font color="Maroon" title="c12">rho</font>,<font color="Maroon" title="c13">D</font>,<font color="Maroon" title="c24">k</font>)</span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c34">f1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c18">t</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c35">f2</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c18">t</font></span>)</span></span>)</span>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="22_2_1_3_1_2_7"><span class="kw">per </span></a><a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:22_2_1_3_1_2_7"/>
( <font color="Maroon" title="c21">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c24">k</font> or <font color="Maroon" title="c24">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c21">i</font> )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:22_2_1_3"><span class="lab"><font color="Green" title="E54">A121</font></span></a>, <a class="ref" href="xxreal_0.html#T1" onmouseover="rs('xxreal_0/T1')" onmouseout="rh()">XXREAL_0:1</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="22_2_1_3_1_2_7_1"><span class="kw">suppose </span></a><a NAME="E1:22_2_1_3_1_2_7_1"/>
<font color="Maroon" title="c21">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c24">k</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="c19">z</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c24">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> </span><br/><div class="add"><a NAME="E2:22_2_1_3_1_2_7_1"/><span class="kw">then </span>
<font color="Maroon" title="c21">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c24">k</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T13" onmouseover="rs('nat_1/T13')" onmouseout="rh()">NAT_1:13</a></span>;<br/><a NAME="E3:22_2_1_3_1_2_7_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E80">A186</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c21">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c24">k</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T13" onmouseover="rs('xreal_1/T13')" onmouseout="rh()">XREAL_1:13</a></span>;<br/><a NAME="E4:22_2_1_3_1_2_7_1"/><span class="lab"><font color="Green" title="E81">A187</font></span>: 
<font color="Maroon" title="c13">D</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c13">D</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c24">k</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span>
 <div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="22_2_1_3_1_2_7_1_1"><span class="kw">proof </span></a><div class="add">

<a NAME="E1:22_2_1_3_1_2_7_1_1"/>
( <font color="Maroon" title="c21">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">k</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 or <font color="Maroon" title="c21">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c24">k</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:22_2_1_3_1_2_7_1"><span class="lab"><font color="Green" title="E80">A186</font></span></a>, <a class="ref" href="xxreal_0.html#T1" onmouseover="rs('xxreal_0/T1')" onmouseout="rh()">XXREAL_0:1</a></span>;<br/>
<span class="kw">hence </span><a NAME="E2:22_2_1_3_1_2_7_1_1"/>
<font color="Maroon" title="c13">D</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c13">D</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c24">k</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:22_2_1"><span class="lab"><font color="Green" title="E41">A50</font></span></a>, <a class="txt" href="#E4:22_2_1_3_1_2"><span class="lab"><font color="Green" title="E67">A168</font></span></a>, <a class="ref" href="valued_0.html#D13" onmouseover="rs('valued_0/D13')" onmouseout="rh()">VALUED_0:def 13</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="E5:22_2_1_3_1_2_7_1"/><span class="lab"><font color="Green" title="E82">A189</font></span>: 
 <a href="seq_4.html#K2" title="SEQ_4:func.2">upper_bound</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c13">D</font>,<font color="Maroon" title="c21">i</font>)</span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c13">D</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c24">k</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span>
 <div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="22_2_1_3_1_2_7_1_2"><span class="kw">proof </span></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="22_2_1_3_1_2_7_1_2_1"><span class="kw">per </span></a><a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:22_2_1_3_1_2_7_1_2_1"/>
( <font color="Maroon" title="c21">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or <font color="Maroon" title="c21">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 1 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="22_2_1_3_1_2_7_1_2_1_1"><span class="kw">suppose </span></a><a NAME="E1:22_2_1_3_1_2_7_1_2_1_1"/>
<font color="Maroon" title="c21">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 ; <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="seq_4.html#K2" title="SEQ_4:func.2">upper_bound</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c13">D</font>,<font color="Maroon" title="c21">i</font>)</span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c13">D</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c24">k</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span></span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:22_2_1_3_1_2_7_1_2_1_1"/>
 <a href="seq_4.html#K2" title="SEQ_4:func.2">upper_bound</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c13">D</font>,<font color="Maroon" title="c21">i</font>)</span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c13">D</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c24">k</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:22_2_1_3_1_2_7_1"><span class="lab"><font color="Green" title="E81">A187</font></span></a>, <a class="txt" href="#E9:22_2_1"><span class="lab"><font color="Green" title="E41">A50</font></span></a>, <a class="ref" href="integra1.html#D4" onmouseover="rs('integra1/D4')" onmouseout="rh()">INTEGRA1:def 4</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><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="22_2_1_3_1_2_7_1_2_1_2"><span class="kw">suppose </span></a><a NAME="E1:22_2_1_3_1_2_7_1_2_1_2"/>
<font color="Maroon" title="c21">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 1
 ; <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="seq_4.html#K2" title="SEQ_4:func.2">upper_bound</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c13">D</font>,<font color="Maroon" title="c21">i</font>)</span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c13">D</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c24">k</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span></span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:22_2_1_3_1_2_7_1_2_1_2"/>
 <a href="seq_4.html#K2" title="SEQ_4:func.2">upper_bound</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c13">D</font>,<font color="Maroon" title="c21">i</font>)</span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c13">D</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c24">k</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:22_2_1_3_1_2_7_1"><span class="lab"><font color="Green" title="E81">A187</font></span></a>, <a class="txt" href="#E9:22_2_1"><span class="lab"><font color="Green" title="E41">A50</font></span></a>, <a class="ref" href="integra1.html#D4" onmouseover="rs('integra1/D4')" onmouseout="rh()">INTEGRA1:def 4</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></div><span class="kw">end;</span></div>

</div><span class="kw">end;</span></div><a NAME="E6:22_2_1_3_1_2_7_1"/><span class="lab"><font color="Green" title="E83">A191</font></span>: 
( <font color="Maroon" title="c9">a</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c18">t</font> &amp; <font color="Maroon" title="c18">t</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c13">D</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c24">k</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:22_2_1_3_1_2_7_1"><span class="lab"><font color="Green" title="E82">A189</font></span></a>, <a class="ref" href="xxreal_0.html#T2" onmouseover="rs('xxreal_0/T2')" onmouseout="rh()">XXREAL_0:2</a>, <a class="txt" href="#E13:22_2_1"><span class="lab"><font color="Green" title="E44">A53</font></span></a>, <a class="ref" href="xxreal_1.html#T1" onmouseover="rs('xxreal_1/T1')" onmouseout="rh()">XXREAL_1:1</a>, <a class="txt" href="#E11:22"><span class="lab"><font color="Green" title="E28">A7</font></span></a></span>;<br/><a NAME="E7:22_2_1_3_1_2_7_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E84">A192</font></span>: 
<font color="Maroon" title="c18">t</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c9">a</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c13">D</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c24">k</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span></span>)</span></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span>
 ;<br/><a NAME="E8:22_2_1_3_1_2_7_1"/>
( <font color="Maroon" title="c9">a</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c18">t</font> &amp; <font color="Maroon" title="c18">t</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c13">D</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c24">k</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:22_2_1_3_1_2"><span class="lab"><font color="Green" title="E68">A169</font></span></a>, <a class="txt" href="#E6:22_2_1_3_1_2_7_1"><span class="lab"><font color="Green" title="E83">A191</font></span></a>, <a class="ref" href="xxreal_0.html#T2" onmouseover="rs('xxreal_0/T2')" onmouseout="rh()">XXREAL_0:2</a></span>;<br/><a NAME="E9:22_2_1_3_1_2_7_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E85">A193</font></span>: 
<font color="Maroon" title="c18">t</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c9">a</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c13">D</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c24">k</font></span>)</span></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span>
 ;<br/><span class="kw">then </span><span class="lab"><font color="Green" title="E86">A196</font></span>: <font color="Maroon" title="c34">f1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c18">t</font> = 
<span class="p1">(<span class="default"><span class="p2"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c9">a</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c13">D</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c24">k</font></span>)</span></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> 1</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">t</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_4.html#T16" onmouseover="rs('funct_4/T16')" onmouseout="rh()">FUNCT_4:16</a>, <a class="txt" href="#E14:22_2_1_3"><span class="lab"><font color="Green" title="E63">B10</font></span></a>, <a class="ref" href="xxreal_1.html#T90" onmouseover="rs('xxreal_1/T90')" onmouseout="rh()">XXREAL_1:90</a></span>
<br/>.= 
1
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:22_2_1_3_1_2_7_1"><span class="lab"><font color="Green" title="E85">A193</font></span></a>, <a class="ref" href="funcop_1.html#T7" onmouseover="rs('funcop_1/T7')" onmouseout="rh()">FUNCOP_1:7</a></span>
;<br/><font color="Maroon" title="c35">f2</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c18">t</font> = 
<span class="p1">(<span class="default"><span class="p2"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c9">a</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c13">D</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c24">k</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span></span>)</span></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> 1</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">t</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:22_2_1_3"><span class="lab"><font color="Green" title="E63">B10</font></span></a>, <a class="txt" href="#E7:22_2_1_3_1_2_7_1"><span class="lab"><font color="Green" title="E84">A192</font></span></a>, <a class="ref" href="funct_4.html#T16" onmouseover="rs('funct_4/T16')" onmouseout="rh()">FUNCT_4:16</a>, <a class="ref" href="xxreal_1.html#T90" onmouseover="rs('xxreal_1/T90')" onmouseout="rh()">XXREAL_1:90</a></span>
<br/>.= 
1
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:22_2_1_3_1_2_7_1"><span class="lab"><font color="Green" title="E84">A192</font></span></a>, <a class="ref" href="funcop_1.html#T7" onmouseover="rs('funcop_1/T7')" onmouseout="rh()">FUNCOP_1:7</a></span>
;<br/><span class="kw">hence </span><a NAME="E12:22_2_1_3_1_2_7_1"/>
<font color="Maroon" title="c19">z</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c24">k</font> <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="txt" href="#E4:22_2_1_3"><span class="lab"><font color="Green" title="E55">A124</font></span></a>, <a class="txt" href="#E24:22_2_1_3_1_2"><span class="lab"><font color="Green" title="E79">A183</font></span></a>, <a class="txt" href="#E10:22_2_1_3_1_2_7_1"><span class="lab"><font color="Green" title="E86">A196</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><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="22_2_1_3_1_2_7_2"><span class="kw">suppose </span></a><a NAME="E1:22_2_1_3_1_2_7_2"/><span class="lab"><font color="Green" title="E80">A198</font></span>: 
<font color="Maroon" title="c24">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c21">i</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="c19">z</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c24">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> </span><br/><div class="add"><a NAME="E2:22_2_1_3_1_2_7_2"/><span class="kw">then </span>
<font color="Maroon" title="c24">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c21">i</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T13" onmouseover="rs('nat_1/T13')" onmouseout="rh()">NAT_1:13</a></span>;<br/><a NAME="E3:22_2_1_3_1_2_7_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E81">A201</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c24">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c21">i</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T13" onmouseover="rs('xreal_1/T13')" onmouseout="rh()">XREAL_1:13</a></span>;<br/><a NAME="E4:22_2_1_3_1_2_7_2"/><span class="lab"><font color="Green" title="E82">A202</font></span>: 
<font color="Maroon" title="c21">i</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c13">D</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:22_2_1"><span class="lab"><font color="Green" title="E41">A50</font></span></a>, <a class="txt" href="#E1:22_2_1_3_1_2_7_2"><span class="lab"><font color="Green" title="E80">A198</font></span></a>, <a class="txt" href="#E8:22_2_1_3"><span class="lab"><font color="Green" title="E59">A127</font></span></a>, <a class="ref" href="integra1.html#T7" onmouseover="rs('integra1/T7')" onmouseout="rh()">INTEGRA1:7</a></span>;<br/><a NAME="E5:22_2_1_3_1_2_7_2"/><span class="lab"><font color="Green" title="E83">A203</font></span>: 
(  <a href="seq_4.html#K3" title="SEQ_4:func.3">lower_bound</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c13">D</font>,<font color="Maroon" title="c21">i</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">D</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c21">i</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span> &amp;  <a href="seq_4.html#K2" title="SEQ_4:func.2">upper_bound</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c13">D</font>,<font color="Maroon" title="c21">i</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">D</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">i</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:22_2_1"><span class="lab"><font color="Green" title="E41">A50</font></span></a>, <a class="txt" href="#E1:22_2_1_3_1_2_7_2"><span class="lab"><font color="Green" title="E80">A198</font></span></a>, <a class="txt" href="#E8:22_2_1_3"><span class="lab"><font color="Green" title="E59">A127</font></span></a>, <a class="ref" href="integra1.html#D4" onmouseover="rs('integra1/D4')" onmouseout="rh()">INTEGRA1:def 4</a></span>;<br/><a NAME="E6:22_2_1_3_1_2_7_2"/>
<font color="Maroon" title="c13">D</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c24">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c13">D</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c21">i</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span>
 <div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="22_2_1_3_1_2_7_2_1"><span class="kw">proof </span></a><div class="add">

<a NAME="E1:22_2_1_3_1_2_7_2_1"/>
( <font color="Maroon" title="c24">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">i</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 or <font color="Maroon" title="c24">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c21">i</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:22_2_1_3_1_2_7_2"><span class="lab"><font color="Green" title="E81">A201</font></span></a>, <a class="ref" href="xxreal_0.html#T1" onmouseover="rs('xxreal_0/T1')" onmouseout="rh()">XXREAL_0:1</a></span>;<br/>
<span class="kw">hence </span><a NAME="E2:22_2_1_3_1_2_7_2_1"/>
<font color="Maroon" title="c13">D</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c24">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c13">D</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c21">i</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:22_2_1_3"><span class="lab"><font color="Green" title="E58">A126</font></span></a>, <a class="txt" href="#E4:22_2_1_3_1_2_7_2"><span class="lab"><font color="Green" title="E82">A202</font></span></a>, <a class="ref" href="valued_0.html#D13" onmouseover="rs('valued_0/D13')" onmouseout="rh()">VALUED_0:def 13</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="E7:22_2_1_3_1_2_7_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E84">A206</font></span>: 
<font color="Maroon" title="c13">D</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c24">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c18">t</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:22_2_1_3_1_2_7_2"><span class="lab"><font color="Green" title="E80">A198</font></span></a>, <a class="txt" href="#E11:22_2_1"><span class="lab"><font color="Green" title="E43">A52</font></span></a>, <a class="txt" href="#E3:22_2_1"><span class="lab"><font color="Green" title="E38">A18</font></span></a>, <a class="txt" href="#E1:22_2_1_3"><span class="lab"><font color="Green" title="E53">A120</font></span></a>, <a class="ref" href="finseq_1.html#T1" onmouseover="rs('finseq_1/T1')" onmouseout="rh()">FINSEQ_1:1</a>, <a class="ref" href="xxreal_0.html#T2" onmouseover="rs('xxreal_0/T2')" onmouseout="rh()">XXREAL_0:2</a>, <a class="txt" href="#E5:22_2_1_3_1_2_7_2"><span class="lab"><font color="Green" title="E83">A203</font></span></a></span>;<br/><a NAME="E8:22_2_1_3_1_2_7_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E85">A207</font></span>: 
<font color="Maroon" title="c13">D</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c24">k</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c18">t</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:22_2_1_3_1_2"><span class="lab"><font color="Green" title="E68">A169</font></span></a>, <a class="ref" href="xxreal_0.html#T2" onmouseover="rs('xxreal_0/T2')" onmouseout="rh()">XXREAL_0:2</a></span>;<br/><a NAME="E9:22_2_1_3_1_2_7_2"/><span class="lab"><font color="Green" title="E86">A208</font></span>: 
( <font color="Maroon" title="c9">a</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c18">t</font> &amp; <font color="Maroon" title="c18">t</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c10">b</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_1.html#T1" onmouseover="rs('xxreal_1/T1')" onmouseout="rh()">XXREAL_1:1</a>, <a class="txt" href="#E11:22"><span class="lab"><font color="Green" title="E28">A7</font></span></a></span>;<br/><a NAME="E10:22_2_1_3_1_2_7_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E87">A209</font></span>: 
<font color="Maroon" title="c18">t</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"><a href="rcomp_1.html#K4" title="RCOMP_1:func.4">].</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c13">D</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c24">k</font></span>)</span>,<font color="Maroon" title="c10">b</font></span><a href="rcomp_1.html#K4" title="RCOMP_1:func.4">.]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:22_2_1_3_1_2_7_2"><span class="lab"><font color="Green" title="E84">A206</font></span></a></span>;<br/><span class="kw">then </span><span class="lab"><font color="Green" title="E88">A211</font></span>: <font color="Maroon" title="c34">f1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c18">t</font> = 
<span class="p1">(<span class="default"><span class="p2"><a href="rcomp_1.html#K4" title="RCOMP_1:func.4">].</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c13">D</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c24">k</font></span>)</span>,<font color="Maroon" title="c10">b</font></span><a href="rcomp_1.html#K4" title="RCOMP_1:func.4">.]</a></span> <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">t</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:22_2_1_3"><span class="lab"><font color="Green" title="E63">B10</font></span></a>, <a class="ref" href="funct_4.html#T13" onmouseover="rs('funct_4/T13')" onmouseout="rh()">FUNCT_4:13</a></span>
<br/>.= 
 <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:22_2_1_3_1_2_7_2"><span class="lab"><font color="Green" title="E87">A209</font></span></a>, <a class="ref" href="funcop_1.html#T7" onmouseover="rs('funcop_1/T7')" onmouseout="rh()">FUNCOP_1:7</a></span>
;<br/><a NAME="E12:22_2_1_3_1_2_7_2"/><span class="lab"><font color="Green" title="E89">A210</font></span>: 
<font color="Maroon" title="c18">t</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"><a href="rcomp_1.html#K4" title="RCOMP_1:func.4">].</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c13">D</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c24">k</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c10">b</font></span><a href="rcomp_1.html#K4" title="RCOMP_1:func.4">.]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:22_2_1_3_1_2_7_2"><span class="lab"><font color="Green" title="E85">A207</font></span></a>, <a class="txt" href="#E9:22_2_1_3_1_2_7_2"><span class="lab"><font color="Green" title="E86">A208</font></span></a></span>;<br/><span class="kw">then </span><font color="Maroon" title="c35">f2</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c18">t</font> = 
<span class="p1">(<span class="default"><span class="p2"><a href="rcomp_1.html#K4" title="RCOMP_1:func.4">].</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c13">D</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c24">k</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c10">b</font></span><a href="rcomp_1.html#K4" title="RCOMP_1:func.4">.]</a></span> <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">t</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:22_2_1_3"><span class="lab"><font color="Green" title="E63">B10</font></span></a>, <a class="ref" href="funct_4.html#T13" onmouseover="rs('funct_4/T13')" onmouseout="rh()">FUNCT_4:13</a></span>
<br/>.= 
 <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:22_2_1_3_1_2_7_2"><span class="lab"><font color="Green" title="E89">A210</font></span></a>, <a class="ref" href="funcop_1.html#T7" onmouseover="rs('funcop_1/T7')" onmouseout="rh()">FUNCOP_1:7</a></span>
;<br/><span class="kw">hence </span><a NAME="E14:22_2_1_3_1_2_7_2"/>
<font color="Maroon" title="c19">z</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c24">k</font> <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="txt" href="#E4:22_2_1_3"><span class="lab"><font color="Green" title="E55">A124</font></span></a>, <a class="txt" href="#E24:22_2_1_3_1_2"><span class="lab"><font color="Green" title="E79">A183</font></span></a>, <a class="txt" href="#E11:22_2_1_3_1_2_7_2"><span class="lab"><font color="Green" title="E88">A211</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></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

</div><span class="kw">end;</span></div>
<a NAME="E29:22_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> <font color="Maroon" title="c19">z</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> 1
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:22_2_1"><span class="lab"><font color="Green" title="E40">A22</font></span></a>, <a class="txt" href="#E9:22_2_1"><span class="lab"><font color="Green" title="E41">A50</font></span></a>, <a class="txt" href="#E27:22_2_1"><span class="lab"><font color="Green" title="E52">A119</font></span></a>, <a class="ref" href="integr23.html#T6" onmouseover="rs('integr23/T6')" onmouseout="rh()">INTEGR23:6</a></span>;<br/>
<span class="kw">hence </span><a NAME="E30:22_2_1"/>
<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">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c18">t</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> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:22_2"><span class="lab"><font color="Green" title="E34">A13</font></span></a>, <a class="txt" href="#E4:22_2_1"><span class="lab"><font color="Green" title="E39">A20</font></span></a>, <a class="ref" href="dualsp05.html#T12" target="_self" onmouseover="rs('dualsp05/T12')" onmouseout="rh()">LM89</a>, <a class="txt" href="#E3:22_2_1"><span class="lab"><font color="Green" title="E38">A18</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>
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="22_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">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="c0sp1.html#K9" title="C0SP1:func.9">PreNorms</a> <font color="Maroon" title="c17">g</font> holds <br/><font color="Olive" title="b1">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1</span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c18">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="c18">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="c0sp1.html#K9" title="C0SP1:func.9">PreNorms</a> <font color="Maroon" title="c17">g</font> implies <font color="Maroon" title="c18">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 )</span><br/><span class="kw">assume </span><a NAME="E1:22_2_2"/>
<font color="Maroon" title="c18">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="c0sp1.html#K9" title="C0SP1:func.9">PreNorms</a> <font color="Maroon" title="c17">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="c18">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1</span><br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c19">t</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c1">A</font><span class="kw"> such that </span><br/><a NAME="E3:22_2_2"/><span class="lab"><font color="Green" title="E37">A213</font></span>: 
<font color="Maroon" title="c18">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="c17">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c19">t</font></span>)</span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span>
 ;<br/><span class="kw">thus </span><a NAME="E4:22_2_2"/>
<font color="Maroon" title="c18">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:22_2"><span class="lab"><font color="Green" title="E36">A15</font></span></a>, <a class="txt" href="#E3:22_2_2"><span class="lab"><font color="Green" title="E37">A213</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="E12:22_2"/><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="c0sp1.html#K9" title="C0SP1:func.9">PreNorms</a> <font color="Maroon" title="c17">g</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1
 
<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/>
<a NAME="E13:22_2"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="c0sp1.html#K10" title="C0SP1:func.10">BoundedFunctionsNorm</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">g</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:22_2"><span class="lab"><font color="Green" title="E34">A13</font></span></a>, <a class="ref" href="c0sp1.html#T20" onmouseover="rs('c0sp1/T20')" onmouseout="rh()">C0SP1:20</a></span>;<br/>
<a NAME="E14:22_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E37">A214</font></span>: 
<span class="p1"><a href="normsp_0.html#K2" title="NORMSP_0:func.2">||.</a><span class="default"><span class="p2">(<span class="default"><a href="rlvect_1.html#K4" title="RLVECT_1:func.4">Sum</a> <font color="Maroon" title="c15">s</font></span>)</span></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
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:22_2"><span class="lab"><font color="Green" title="E34">A13</font></span></a>, <a class="ref" href="dualsp05.html#T5" target="_self" onmouseover="rs('dualsp05/T5')" onmouseout="rh()">Lm1</a></span>;<br/>
<a NAME="E15:22_2"/><span class="lab"><font color="Green" title="E38">A215</font></span>: 
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c14">K</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c13">D</font> &amp; (  for <font color="Olive" title="b1">i</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Olive" title="b1">i</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> <font color="Maroon" title="c13">D</font> holds <br/><font color="Maroon" title="c14">K</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</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"><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> (<font color="Maroon" title="c13">D</font>,<font color="Olive" title="b1">i</font>)</span>)</span>,<font color="Maroon" title="c12">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="integr22.html#D2" onmouseover="rs('integr22/D2')" onmouseout="rh()">INTEGR22:def 2</a></span>;<br/>
<a NAME="E16:22_2"/>
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c7">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="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="funct_2.html#D1" onmouseover="rs('funct_2/D1')" onmouseout="rh()">FUNCT_2:def 1</a></span>;<br/>
<a NAME="E17:22_2"/><span class="kw">then </span>
 <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c15">s</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c7">f</font>
 
;<br/>
<span class="kw">then </span><span class="lab"><font color="Green" title="E39">A216</font></span>:  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">f</font> <a href="finseqop.html#K4" title="FINSEQOP:func.4">*</a> <font color="Maroon" title="c15">s</font></span>)</span> = 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c15">s</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#T27" onmouseover="rs('relat_1/T27')" onmouseout="rh()">RELAT_1:27</a></span>
<br/>.= 
 <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> <font color="Maroon" title="c15">s</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 href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c14">K</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:22_2"><span class="lab"><font color="Green" title="E32">A11</font></span></a>, <a class="txt" href="#E15:22_2"><span class="lab"><font color="Green" title="E38">A215</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="E19:22_2"/><span class="lab"><font color="Green" title="E40">A217</font></span>: 
 for <font color="Olive" title="b1">i</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Olive" title="b1">i</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> <font color="Maroon" title="c13">D</font> holds <br/><span class="p1">(<span class="default"><font color="Maroon" title="c7">f</font> <a href="finseqop.html#K4" title="FINSEQOP:func.4">*</a> <font color="Maroon" title="c15">s</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</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"><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> (<font color="Maroon" title="c13">D</font>,<font color="Olive" title="b1">i</font>)</span>)</span>,<font color="Maroon" title="c12">rho</font>)</span>)</span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="22_2_4"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c18">i</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="c18">i</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> <font color="Maroon" title="c13">D</font> implies <span class="p1">(<span class="default"><font color="Maroon" title="c7">f</font> <a href="finseqop.html#K4" title="FINSEQOP:func.4">*</a> <font color="Maroon" title="c15">s</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">i</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"><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> (<font color="Maroon" title="c13">D</font>,<font color="Maroon" title="c18">i</font>)</span>)</span>,<font color="Maroon" title="c12">rho</font>)</span>)</span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span> )</span><br/>

<span class="kw">assume </span><a NAME="E1:22_2_4"/><span class="lab"><font color="Green" title="E41">A218</font></span>: 
<font color="Maroon" title="c18">i</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> <font color="Maroon" title="c13">D</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">(<span class="default"><font color="Maroon" title="c7">f</font> <a href="finseqop.html#K4" title="FINSEQOP:func.4">*</a> <font color="Maroon" title="c15">s</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">i</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"><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> (<font color="Maroon" title="c13">D</font>,<font color="Maroon" title="c18">i</font>)</span>)</span>,<font color="Maroon" title="c12">rho</font>)</span>)</span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span></span><br/>

<a NAME="E2:22_2_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E42">A220</font></span>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c18">i</font> &amp; <font color="Maroon" title="c18">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c13">D</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_3.html#T25" onmouseover="rs('finseq_3/T25')" onmouseout="rh()">FINSEQ_3:25</a></span>;<br/>
<a NAME="E3:22_2_4"/><span class="kw">then </span>
<font color="Maroon" title="c18">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c13">D</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T7" onmouseover="rs('xreal_1/T7')" onmouseout="rh()">XREAL_1:7</a></span>;<br/>
<a NAME="E4:22_2_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E43">A221</font></span>: 
<font color="Maroon" title="c18">i</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"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c13">D</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:22_2_4"><span class="lab"><font color="Green" title="E42">A220</font></span></a></span>;<br/>
<a NAME="E5:22_2_4"/><span class="lab"><font color="Green" title="E44">b</font></span>: 
1 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c18">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T7" onmouseover="rs('xreal_1/T7')" onmouseout="rh()">XREAL_1:7</a></span>;<br/>
<a NAME="E6:22_2_4"/>
<font color="Maroon" title="c18">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c13">D</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:22_2_4"><span class="lab"><font color="Green" title="E42">A220</font></span></a>, <a class="ref" href="xreal_1.html#T7" onmouseover="rs('xreal_1/T7')" onmouseout="rh()">XREAL_1:7</a></span>;<br/>
<a NAME="E7:22_2_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E45">A222</font></span>: 
<font color="Maroon" title="c18">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <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"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c13">D</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:22_2_4"><span class="lab"><font color="Green" title="E44">b</font></span></a></span>;<br/>
<a NAME="E8:22_2_4"/>
<font color="Maroon" title="c18">i</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> <font color="Maroon" title="c15">s</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:22_2"><span class="lab"><font color="Green" title="E32">A11</font></span></a>, <a class="txt" href="#E1:22_2_4"><span class="lab"><font color="Green" title="E41">A218</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="E9:22_2_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E46">A223</font></span>: 
<font color="Maroon" title="c18">i</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> <font color="Maroon" title="c15">s</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/>
<span class="kw">set </span><font color="Maroon" title="c19">r</font> =  <a href="absvalue.html#K1" title="ABSVALUE:func.1">sgn</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="dualsp05.html#K3" title="DUALSP05:func.3">Dp2</a> (<font color="Maroon" title="c12">rho</font>,<font color="Maroon" title="c13">D</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c18">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>)</span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p2">(<span class="default"><a href="dualsp05.html#K3" title="DUALSP05:func.3">Dp2</a> (<font color="Maroon" title="c12">rho</font>,<font color="Maroon" title="c13">D</font>,<font color="Maroon" title="c18">i</font>)</span>)</span></span>)</span>;<br/>
<a NAME="E10:22_2_4"/>
<font color="Maroon" title="c1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><span class="p2">(<span class="default"><a href="seq_4.html#K3" title="SEQ_4:func.3">lower_bound</a> <font color="Maroon" title="c1">A</font></span>)</span>,<span class="p2">(<span class="default"><a href="seq_4.html#K2" title="SEQ_4:func.2">upper_bound</a> <font color="Maroon" title="c1">A</font></span>)</span></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="integra1.html#T4" onmouseover="rs('integra1/T4')" onmouseout="rh()">INTEGRA1:4</a></span>;<br/>
<a NAME="E11:22_2_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E47">A224</font></span>: 
 <a href="seq_4.html#K3" title="SEQ_4:func.3">lower_bound</a> <font color="Maroon" title="c1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">a</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:22"><span class="lab"><font color="Green" title="E28">A7</font></span></a>, <a class="ref" href="integra1.html#T5" onmouseover="rs('integra1/T5')" onmouseout="rh()">INTEGRA1:5</a></span>;<br/>
<a NAME="E12:22_2_4"/>
<font color="Maroon" title="c13">D</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">i</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c1">A</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:22_2_4"><span class="lab"><font color="Green" title="E41">A218</font></span></a>, <a class="ref" href="integra1.html#T6" onmouseover="rs('integra1/T6')" onmouseout="rh()">INTEGRA1:6</a></span>;<br/>
<a NAME="E13:22_2_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E48">A225</font></span>: 
<font color="Maroon" title="c13">D</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">i</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c11">m</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D1" onmouseover="rs('funct_2/D1')" onmouseout="rh()">FUNCT_2:def 1</a></span>;<br/>
<a NAME="E14:22_2_4"/>
 <a href="seq_4.html#K3" title="SEQ_4:func.3">lower_bound</a> <font color="Maroon" title="c1">A</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="#E11:22"><span class="lab"><font color="Green" title="E28">A7</font></span></a>, <a class="txt" href="#E11:22_2_4"><span class="lab"><font color="Green" title="E47">A224</font></span></a></span>;<br/>
<a NAME="E15:22_2_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E49">A226</font></span>: 
 <a href="seq_4.html#K3" title="SEQ_4:func.3">lower_bound</a> <font color="Maroon" title="c1">A</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> <font color="Maroon" title="c11">m</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D1" onmouseover="rs('funct_2/D1')" onmouseout="rh()">FUNCT_2:def 1</a></span>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="22_2_4_1"><span class="kw">per </span></a><a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:22_2_4_1"/>
( <font color="Maroon" title="c18">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or <font color="Maroon" title="c18">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 1 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="22_2_4_1_1"><span class="kw">suppose </span></a><a NAME="E1:22_2_4_1_1"/><span class="lab"><font color="Green" title="E50">A227</font></span>: 
<font color="Maroon" title="c18">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c7">f</font> <a href="finseqop.html#K4" title="FINSEQOP:func.4">*</a> <font color="Maroon" title="c15">s</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">i</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"><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> (<font color="Maroon" title="c13">D</font>,<font color="Maroon" title="c18">i</font>)</span>)</span>,<font color="Maroon" title="c12">rho</font>)</span>)</span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span></span><br/><div class="add"><a NAME="E2:22_2_4_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E51">A228</font></span>: 
(  <a href="seq_4.html#K3" title="SEQ_4:func.3">lower_bound</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c13">D</font>,<font color="Maroon" title="c18">i</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="seq_4.html#K3" title="SEQ_4:func.3">lower_bound</a> <font color="Maroon" title="c1">A</font> &amp;  <a href="seq_4.html#K2" title="SEQ_4:func.2">upper_bound</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c13">D</font>,<font color="Maroon" title="c18">i</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">D</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">i</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:22_2_4"><span class="lab"><font color="Green" title="E41">A218</font></span></a>, <a class="ref" href="integra1.html#D4" onmouseover="rs('integra1/D4')" onmouseout="rh()">INTEGRA1:def 4</a></span>;<br/><span class="lab"><font color="Green" title="E52">A229</font></span>:  <a href="dualsp05.html#K2" title="DUALSP05:func.2">Dp1</a> (<font color="Maroon" title="c11">m</font>,<font color="Maroon" title="c13">D</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c18">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>) = 
<font color="Maroon" title="c11">m</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">D</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c18">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:22_2_4"><span class="lab"><font color="Green" title="E45">A222</font></span></a>, <a class="ref" href="dualsp05.html#D2" target="_self" onmouseover="rs('dualsp05/D2')" onmouseout="rh()">defDp1</a>, <a class="txt" href="#E1:22_2_4_1_1"><span class="lab"><font color="Green" title="E50">A227</font></span></a></span>
<br/>.= 
<font color="Maroon" title="c11">m</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">D</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">i</font></span>)</span>

;<br/><span class="lab"><font color="Green" title="E53">A231</font></span>: <span class="p1">(<span class="default"><font color="Maroon" title="c7">f</font> <a href="finseqop.html#K4" title="FINSEQOP:func.4">*</a> <font color="Maroon" title="c15">s</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">i</font> = 
<font color="Maroon" title="c7">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">s</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">i</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:22_2_4"><span class="lab"><font color="Green" title="E46">A223</font></span></a>, <a class="ref" href="funct_1.html#T13" onmouseover="rs('funct_1/T13')" onmouseout="rh()">FUNCT_1:13</a></span>
<br/>.= 
<font color="Maroon" title="c7">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="absvalue.html#K1" title="ABSVALUE:func.1">sgn</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="dualsp05.html#K3" title="DUALSP05:func.3">Dp2</a> (<font color="Maroon" title="c12">rho</font>,<font color="Maroon" title="c13">D</font>,<span class="p5">(<span class="default"><font color="Maroon" title="c18">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>)</span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p4">(<span class="default"><a href="dualsp05.html#K3" title="DUALSP05:func.3">Dp2</a> (<font color="Maroon" title="c12">rho</font>,<font color="Maroon" title="c13">D</font>,<font color="Maroon" title="c18">i</font>)</span>)</span></span>)</span></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="dualsp05.html#K2" title="DUALSP05:func.2">Dp1</a> (<font color="Maroon" title="c11">m</font>,<font color="Maroon" title="c13">D</font>,<span class="p4">(<span class="default"><font color="Maroon" title="c18">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>)</span>)</span> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <span class="p3">(<span class="default"><a href="dualsp05.html#K2" title="DUALSP05:func.2">Dp1</a> (<font color="Maroon" title="c11">m</font>,<font color="Maroon" title="c13">D</font>,<font color="Maroon" title="c18">i</font>)</span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:22_2"><span class="lab"><font color="Green" title="E33">A12</font></span></a>, <a class="txt" href="#E9:22_2_4"><span class="lab"><font color="Green" title="E46">A223</font></span></a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="absvalue.html#K1" title="ABSVALUE:func.1">sgn</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="dualsp05.html#K3" title="DUALSP05:func.3">Dp2</a> (<font color="Maroon" title="c12">rho</font>,<font color="Maroon" title="c13">D</font>,<span class="p4">(<span class="default"><font color="Maroon" title="c18">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>)</span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p3">(<span class="default"><a href="dualsp05.html#K3" title="DUALSP05:func.3">Dp2</a> (<font color="Maroon" title="c12">rho</font>,<font color="Maroon" title="c13">D</font>,<font color="Maroon" title="c18">i</font>)</span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="dualsp05.html#K2" title="DUALSP05:func.2">Dp1</a> (<font color="Maroon" title="c11">m</font>,<font color="Maroon" title="c13">D</font>,<span class="p4">(<span class="default"><font color="Maroon" title="c18">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>)</span>)</span> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <span class="p3">(<span class="default"><a href="dualsp05.html#K2" title="DUALSP05:func.2">Dp1</a> (<font color="Maroon" title="c11">m</font>,<font color="Maroon" title="c13">D</font>,<font color="Maroon" title="c18">i</font>)</span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="hahnban.html#D3" onmouseover="rs('hahnban/D3')" onmouseout="rh()">HAHNBAN:def 3</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="absvalue.html#K1" title="ABSVALUE:func.1">sgn</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="dualsp05.html#K3" title="DUALSP05:func.3">Dp2</a> (<font color="Maroon" title="c12">rho</font>,<font color="Maroon" title="c13">D</font>,<span class="p4">(<span class="default"><font color="Maroon" title="c18">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>)</span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p3">(<span class="default"><a href="dualsp05.html#K3" title="DUALSP05:func.3">Dp2</a> (<font color="Maroon" title="c12">rho</font>,<font color="Maroon" title="c13">D</font>,<font color="Maroon" title="c18">i</font>)</span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p3">(<span class="default"><a href="dualsp05.html#K2" title="DUALSP05:func.2">Dp1</a> (<font color="Maroon" title="c11">m</font>,<font color="Maroon" title="c13">D</font>,<span class="p4">(<span class="default"><font color="Maroon" title="c18">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>)</span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p3">(<span class="default"><a href="dualsp05.html#K2" title="DUALSP05:func.2">Dp1</a> (<font color="Maroon" title="c11">m</font>,<font color="Maroon" title="c13">D</font>,<font color="Maroon" title="c18">i</font>)</span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="hahnban.html#T19" onmouseover="rs('hahnban/T19')" onmouseout="rh()">HAHNBAN:19</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="absvalue.html#K1" title="ABSVALUE:func.1">sgn</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="dualsp05.html#K3" title="DUALSP05:func.3">Dp2</a> (<font color="Maroon" title="c12">rho</font>,<font color="Maroon" title="c13">D</font>,<span class="p4">(<span class="default"><font color="Maroon" title="c18">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>)</span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p3">(<span class="default"><a href="dualsp05.html#K3" title="DUALSP05:func.3">Dp2</a> (<font color="Maroon" title="c12">rho</font>,<font color="Maroon" title="c13">D</font>,<font color="Maroon" title="c18">i</font>)</span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c11">m</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c13">D</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">i</font></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c11">m</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p4">(<span class="default"><a href="seq_4.html#K3" title="SEQ_4:func.3">lower_bound</a> <font color="Maroon" title="c1">A</font></span>)</span></span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:22_2_4_1_1"><span class="lab"><font color="Green" title="E52">A229</font></span></a>, <a class="txt" href="#E4:22_2_4"><span class="lab"><font color="Green" title="E43">A221</font></span></a>, <a class="txt" href="#E1:22_2_4_1_1"><span class="lab"><font color="Green" title="E50">A227</font></span></a>, <a class="ref" href="dualsp05.html#D2" target="_self" onmouseover="rs('dualsp05/D2')" onmouseout="rh()">defDp1</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="absvalue.html#K1" title="ABSVALUE:func.1">sgn</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="dualsp05.html#K3" title="DUALSP05:func.3">Dp2</a> (<font color="Maroon" title="c12">rho</font>,<font color="Maroon" title="c13">D</font>,<span class="p4">(<span class="default"><font color="Maroon" title="c18">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>)</span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p3">(<span class="default"><a href="dualsp05.html#K3" title="DUALSP05:func.3">Dp2</a> (<font color="Maroon" title="c12">rho</font>,<font color="Maroon" title="c13">D</font>,<font color="Maroon" title="c18">i</font>)</span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c7">f</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c11">m</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c13">D</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">i</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c11">m</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p4">(<span class="default"><a href="seq_4.html#K3" title="SEQ_4:func.3">lower_bound</a> <font color="Maroon" title="c1">A</font></span>)</span></span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:22_2_4"><span class="lab"><font color="Green" title="E48">A225</font></span></a>, <a class="ref" href="funct_1.html#T13" onmouseover="rs('funct_1/T13')" onmouseout="rh()">FUNCT_1:13</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="absvalue.html#K1" title="ABSVALUE:func.1">sgn</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="dualsp05.html#K3" title="DUALSP05:func.3">Dp2</a> (<font color="Maroon" title="c12">rho</font>,<font color="Maroon" title="c13">D</font>,<span class="p4">(<span class="default"><font color="Maroon" title="c18">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>)</span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p3">(<span class="default"><a href="dualsp05.html#K3" title="DUALSP05:func.3">Dp2</a> (<font color="Maroon" title="c12">rho</font>,<font color="Maroon" title="c13">D</font>,<font color="Maroon" title="c18">i</font>)</span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c12">rho</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><a href="seq_4.html#K2" title="SEQ_4:func.2">upper_bound</a> <span class="p4">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c13">D</font>,<font color="Maroon" title="c18">i</font>)</span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c12">rho</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><a href="seq_4.html#K3" title="SEQ_4:func.3">lower_bound</a> <span class="p4">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c13">D</font>,<font color="Maroon" title="c18">i</font>)</span>)</span></span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:22_2_4_1_1"><span class="lab"><font color="Green" title="E51">A228</font></span></a>, <a class="txt" href="#E15:22_2_4"><span class="lab"><font color="Green" title="E49">A226</font></span></a>, <a class="ref" href="funct_1.html#T13" onmouseover="rs('funct_1/T13')" onmouseout="rh()">FUNCT_1:13</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="absvalue.html#K1" title="ABSVALUE:func.1">sgn</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="dualsp05.html#K3" title="DUALSP05:func.3">Dp2</a> (<font color="Maroon" title="c12">rho</font>,<font color="Maroon" title="c13">D</font>,<span class="p4">(<span class="default"><font color="Maroon" title="c18">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>)</span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p3">(<span class="default"><a href="dualsp05.html#K3" title="DUALSP05:func.3">Dp2</a> (<font color="Maroon" title="c12">rho</font>,<font color="Maroon" title="c13">D</font>,<font color="Maroon" title="c18">i</font>)</span>)</span></span>)</span></span>)</span> <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> (<font color="Maroon" title="c13">D</font>,<font color="Maroon" title="c18">i</font>)</span>)</span>,<font color="Maroon" title="c12">rho</font>)</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="integr22.html#D1" onmouseover="rs('integr22/D1')" onmouseout="rh()">INTEGR22:def 1</a></span>
;<br/><span class="lab"><font color="Green" title="E54">A232</font></span>:  <a href="dualsp05.html#K3" title="DUALSP05:func.3">Dp2</a> (<font color="Maroon" title="c12">rho</font>,<font color="Maroon" title="c13">D</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c18">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>) = 
<font color="Maroon" title="c12">rho</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">D</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c18">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="dualsp05.html#D3" target="_self" onmouseover="rs('dualsp05/D3')" onmouseout="rh()">defDp2</a>, <a class="txt" href="#E1:22_2_4_1_1"><span class="lab"><font color="Green" title="E50">A227</font></span></a></span>
<br/>.= 
<font color="Maroon" title="c12">rho</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">D</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">i</font></span>)</span>

;<br/> <a href="absvalue.html#K1" title="ABSVALUE:func.1">sgn</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="dualsp05.html#K3" title="DUALSP05:func.3">Dp2</a> (<font color="Maroon" title="c12">rho</font>,<font color="Maroon" title="c13">D</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c18">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>)</span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p2">(<span class="default"><a href="dualsp05.html#K3" title="DUALSP05:func.3">Dp2</a> (<font color="Maroon" title="c12">rho</font>,<font color="Maroon" title="c13">D</font>,<font color="Maroon" title="c18">i</font>)</span>)</span></span>)</span> = 
 <a href="absvalue.html#K1" title="ABSVALUE:func.1">sgn</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c12">rho</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><a href="seq_4.html#K2" title="SEQ_4:func.2">upper_bound</a> <span class="p4">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c13">D</font>,<font color="Maroon" title="c18">i</font>)</span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c12">rho</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><a href="seq_4.html#K3" title="SEQ_4:func.3">lower_bound</a> <span class="p4">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c13">D</font>,<font color="Maroon" title="c18">i</font>)</span>)</span></span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:22_2_4_1_1"><span class="lab"><font color="Green" title="E51">A228</font></span></a>, <a class="txt" href="#E5:22_2_4_1_1"><span class="lab"><font color="Green" title="E54">A232</font></span></a>, <a class="txt" href="#E1:22_2_4_1_1"><span class="lab"><font color="Green" title="E50">A227</font></span></a>, <a class="ref" href="dualsp05.html#D3" target="_self" onmouseover="rs('dualsp05/D3')" onmouseout="rh()">defDp2</a></span>
<br/>.= 
 <a href="absvalue.html#K1" title="ABSVALUE:func.1">sgn</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> (<font color="Maroon" title="c13">D</font>,<font color="Maroon" title="c18">i</font>)</span>)</span>,<font color="Maroon" title="c12">rho</font>)</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="integr22.html#D1" onmouseover="rs('integr22/D1')" onmouseout="rh()">INTEGR22:def 1</a></span>
;<br/><span class="kw">hence </span><a NAME="E7:22_2_4_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c7">f</font> <a href="finseqop.html#K4" title="FINSEQOP:func.4">*</a> <font color="Maroon" title="c15">s</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">i</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"><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> (<font color="Maroon" title="c13">D</font>,<font color="Maroon" title="c18">i</font>)</span>)</span>,<font color="Maroon" title="c12">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="txt" href="#E4:22_2_4_1_1"><span class="lab"><font color="Green" title="E53">A231</font></span></a>, <a class="ref" href="dualsp05.html#T2" target="_self" onmouseover="rs('dualsp05/T2')" onmouseout="rh()">LM86</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><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="22_2_4_1_2"><span class="kw">suppose </span></a><a NAME="E1:22_2_4_1_2"/><span class="lab"><font color="Green" title="E50">A234</font></span>: 
<font color="Maroon" title="c18">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 1
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c7">f</font> <a href="finseqop.html#K4" title="FINSEQOP:func.4">*</a> <font color="Maroon" title="c15">s</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">i</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"><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> (<font color="Maroon" title="c13">D</font>,<font color="Maroon" title="c18">i</font>)</span>)</span>,<font color="Maroon" title="c12">rho</font>)</span>)</span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span></span><br/><div class="add"><a NAME="E2:22_2_4_1_2"/>
<font color="Maroon" title="c13">D</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">i</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c1">A</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:22_2_4"><span class="lab"><font color="Green" title="E41">A218</font></span></a>, <a class="ref" href="integra1.html#T6" onmouseover="rs('integra1/T6')" onmouseout="rh()">INTEGRA1:6</a></span>;<br/><a NAME="E3:22_2_4_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E51">A235</font></span>: 
<font color="Maroon" title="c13">D</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">i</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c11">m</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D1" onmouseover="rs('funct_2/D1')" onmouseout="rh()">FUNCT_2:def 1</a></span>;<br/><a NAME="E4:22_2_4_1_2"/>
<font color="Maroon" title="c13">D</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c18">i</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c1">A</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:22_2_4"><span class="lab"><font color="Green" title="E41">A218</font></span></a>, <a class="txt" href="#E1:22_2_4_1_2"><span class="lab"><font color="Green" title="E50">A234</font></span></a>, <a class="ref" href="integra1.html#T7" onmouseover="rs('integra1/T7')" onmouseout="rh()">INTEGRA1:7</a></span>;<br/><a NAME="E5:22_2_4_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E52">A236</font></span>: 
<font color="Maroon" title="c13">D</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c18">i</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c11">m</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D1" onmouseover="rs('funct_2/D1')" onmouseout="rh()">FUNCT_2:def 1</a></span>;<br/><a NAME="E6:22_2_4_1_2"/><span class="lab"><font color="Green" title="E53">A237</font></span>: 
(  <a href="seq_4.html#K3" title="SEQ_4:func.3">lower_bound</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c13">D</font>,<font color="Maroon" title="c18">i</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">D</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c18">i</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span> &amp;  <a href="seq_4.html#K2" title="SEQ_4:func.2">upper_bound</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c13">D</font>,<font color="Maroon" title="c18">i</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">D</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">i</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:22_2_4"><span class="lab"><font color="Green" title="E41">A218</font></span></a>, <a class="txt" href="#E1:22_2_4_1_2"><span class="lab"><font color="Green" title="E50">A234</font></span></a>, <a class="ref" href="integra1.html#D4" onmouseover="rs('integra1/D4')" onmouseout="rh()">INTEGRA1:def 4</a></span>;<br/><a NAME="E7:22_2_4_1_2"/><span class="lab"><font color="Green" title="E54">A238</font></span>: 
1 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c18">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T8" onmouseover="rs('xreal_1/T8')" onmouseout="rh()">XREAL_1:8</a>, <a class="txt" href="#E2:22_2_4"><span class="lab"><font color="Green" title="E42">A220</font></span></a></span>;<br/><span class="kw">then </span><span class="lab"><font color="Green" title="E55">A239</font></span>:  <a href="dualsp05.html#K2" title="DUALSP05:func.2">Dp1</a> (<font color="Maroon" title="c11">m</font>,<font color="Maroon" title="c13">D</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c18">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>) = 
<font color="Maroon" title="c11">m</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">D</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c18">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:22_2_4"><span class="lab"><font color="Green" title="E45">A222</font></span></a>, <a class="ref" href="dualsp05.html#D2" target="_self" onmouseover="rs('dualsp05/D2')" onmouseout="rh()">defDp1</a></span>
<br/>.= 
<font color="Maroon" title="c11">m</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">D</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">i</font></span>)</span>

;<br/><span class="lab"><font color="Green" title="E56">A241</font></span>: <span class="p1">(<span class="default"><font color="Maroon" title="c7">f</font> <a href="finseqop.html#K4" title="FINSEQOP:func.4">*</a> <font color="Maroon" title="c15">s</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">i</font> = 
<font color="Maroon" title="c7">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">s</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">i</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:22_2_4"><span class="lab"><font color="Green" title="E46">A223</font></span></a>, <a class="ref" href="funct_1.html#T13" onmouseover="rs('funct_1/T13')" onmouseout="rh()">FUNCT_1:13</a></span>
<br/>.= 
<font color="Maroon" title="c7">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="absvalue.html#K1" title="ABSVALUE:func.1">sgn</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="dualsp05.html#K3" title="DUALSP05:func.3">Dp2</a> (<font color="Maroon" title="c12">rho</font>,<font color="Maroon" title="c13">D</font>,<span class="p5">(<span class="default"><font color="Maroon" title="c18">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>)</span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p4">(<span class="default"><a href="dualsp05.html#K3" title="DUALSP05:func.3">Dp2</a> (<font color="Maroon" title="c12">rho</font>,<font color="Maroon" title="c13">D</font>,<font color="Maroon" title="c18">i</font>)</span>)</span></span>)</span></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="dualsp05.html#K2" title="DUALSP05:func.2">Dp1</a> (<font color="Maroon" title="c11">m</font>,<font color="Maroon" title="c13">D</font>,<span class="p4">(<span class="default"><font color="Maroon" title="c18">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>)</span>)</span> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <span class="p3">(<span class="default"><a href="dualsp05.html#K2" title="DUALSP05:func.2">Dp1</a> (<font color="Maroon" title="c11">m</font>,<font color="Maroon" title="c13">D</font>,<font color="Maroon" title="c18">i</font>)</span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:22_2"><span class="lab"><font color="Green" title="E33">A12</font></span></a>, <a class="txt" href="#E9:22_2_4"><span class="lab"><font color="Green" title="E46">A223</font></span></a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="absvalue.html#K1" title="ABSVALUE:func.1">sgn</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="dualsp05.html#K3" title="DUALSP05:func.3">Dp2</a> (<font color="Maroon" title="c12">rho</font>,<font color="Maroon" title="c13">D</font>,<span class="p4">(<span class="default"><font color="Maroon" title="c18">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>)</span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p3">(<span class="default"><a href="dualsp05.html#K3" title="DUALSP05:func.3">Dp2</a> (<font color="Maroon" title="c12">rho</font>,<font color="Maroon" title="c13">D</font>,<font color="Maroon" title="c18">i</font>)</span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="dualsp05.html#K2" title="DUALSP05:func.2">Dp1</a> (<font color="Maroon" title="c11">m</font>,<font color="Maroon" title="c13">D</font>,<span class="p4">(<span class="default"><font color="Maroon" title="c18">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>)</span>)</span> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <span class="p3">(<span class="default"><a href="dualsp05.html#K2" title="DUALSP05:func.2">Dp1</a> (<font color="Maroon" title="c11">m</font>,<font color="Maroon" title="c13">D</font>,<font color="Maroon" title="c18">i</font>)</span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="hahnban.html#D3" onmouseover="rs('hahnban/D3')" onmouseout="rh()">HAHNBAN:def 3</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="absvalue.html#K1" title="ABSVALUE:func.1">sgn</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="dualsp05.html#K3" title="DUALSP05:func.3">Dp2</a> (<font color="Maroon" title="c12">rho</font>,<font color="Maroon" title="c13">D</font>,<span class="p4">(<span class="default"><font color="Maroon" title="c18">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>)</span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p3">(<span class="default"><a href="dualsp05.html#K3" title="DUALSP05:func.3">Dp2</a> (<font color="Maroon" title="c12">rho</font>,<font color="Maroon" title="c13">D</font>,<font color="Maroon" title="c18">i</font>)</span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p3">(<span class="default"><a href="dualsp05.html#K2" title="DUALSP05:func.2">Dp1</a> (<font color="Maroon" title="c11">m</font>,<font color="Maroon" title="c13">D</font>,<span class="p4">(<span class="default"><font color="Maroon" title="c18">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>)</span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p3">(<span class="default"><a href="dualsp05.html#K2" title="DUALSP05:func.2">Dp1</a> (<font color="Maroon" title="c11">m</font>,<font color="Maroon" title="c13">D</font>,<font color="Maroon" title="c18">i</font>)</span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="hahnban.html#T19" onmouseover="rs('hahnban/T19')" onmouseout="rh()">HAHNBAN:19</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="absvalue.html#K1" title="ABSVALUE:func.1">sgn</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="dualsp05.html#K3" title="DUALSP05:func.3">Dp2</a> (<font color="Maroon" title="c12">rho</font>,<font color="Maroon" title="c13">D</font>,<span class="p4">(<span class="default"><font color="Maroon" title="c18">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>)</span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p3">(<span class="default"><a href="dualsp05.html#K3" title="DUALSP05:func.3">Dp2</a> (<font color="Maroon" title="c12">rho</font>,<font color="Maroon" title="c13">D</font>,<font color="Maroon" title="c18">i</font>)</span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c11">m</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c13">D</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">i</font></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c11">m</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c13">D</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p5">(<span class="default"><font color="Maroon" title="c18">i</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span></span>)</span></span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:22_2_4_1_2"><span class="lab"><font color="Green" title="E55">A239</font></span></a>, <a class="txt" href="#E4:22_2_4"><span class="lab"><font color="Green" title="E43">A221</font></span></a>, <a class="txt" href="#E1:22_2_4_1_2"><span class="lab"><font color="Green" title="E50">A234</font></span></a>, <a class="ref" href="dualsp05.html#D2" target="_self" onmouseover="rs('dualsp05/D2')" onmouseout="rh()">defDp1</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="absvalue.html#K1" title="ABSVALUE:func.1">sgn</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="dualsp05.html#K3" title="DUALSP05:func.3">Dp2</a> (<font color="Maroon" title="c12">rho</font>,<font color="Maroon" title="c13">D</font>,<span class="p4">(<span class="default"><font color="Maroon" title="c18">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>)</span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p3">(<span class="default"><a href="dualsp05.html#K3" title="DUALSP05:func.3">Dp2</a> (<font color="Maroon" title="c12">rho</font>,<font color="Maroon" title="c13">D</font>,<font color="Maroon" title="c18">i</font>)</span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c7">f</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c11">m</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c13">D</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">i</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c11">m</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c13">D</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p5">(<span class="default"><font color="Maroon" title="c18">i</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span></span>)</span></span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:22_2_4_1_2"><span class="lab"><font color="Green" title="E51">A235</font></span></a>, <a class="ref" href="funct_1.html#T13" onmouseover="rs('funct_1/T13')" onmouseout="rh()">FUNCT_1:13</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="absvalue.html#K1" title="ABSVALUE:func.1">sgn</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="dualsp05.html#K3" title="DUALSP05:func.3">Dp2</a> (<font color="Maroon" title="c12">rho</font>,<font color="Maroon" title="c13">D</font>,<span class="p4">(<span class="default"><font color="Maroon" title="c18">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>)</span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p3">(<span class="default"><a href="dualsp05.html#K3" title="DUALSP05:func.3">Dp2</a> (<font color="Maroon" title="c12">rho</font>,<font color="Maroon" title="c13">D</font>,<font color="Maroon" title="c18">i</font>)</span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c12">rho</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><a href="seq_4.html#K2" title="SEQ_4:func.2">upper_bound</a> <span class="p4">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c13">D</font>,<font color="Maroon" title="c18">i</font>)</span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c12">rho</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><a href="seq_4.html#K3" title="SEQ_4:func.3">lower_bound</a> <span class="p4">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c13">D</font>,<font color="Maroon" title="c18">i</font>)</span>)</span></span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:22_2_4_1_2"><span class="lab"><font color="Green" title="E53">A237</font></span></a>, <a class="txt" href="#E5:22_2_4_1_2"><span class="lab"><font color="Green" title="E52">A236</font></span></a>, <a class="ref" href="funct_1.html#T13" onmouseover="rs('funct_1/T13')" onmouseout="rh()">FUNCT_1:13</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="absvalue.html#K1" title="ABSVALUE:func.1">sgn</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="dualsp05.html#K3" title="DUALSP05:func.3">Dp2</a> (<font color="Maroon" title="c12">rho</font>,<font color="Maroon" title="c13">D</font>,<span class="p4">(<span class="default"><font color="Maroon" title="c18">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>)</span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p3">(<span class="default"><a href="dualsp05.html#K3" title="DUALSP05:func.3">Dp2</a> (<font color="Maroon" title="c12">rho</font>,<font color="Maroon" title="c13">D</font>,<font color="Maroon" title="c18">i</font>)</span>)</span></span>)</span></span>)</span> <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> (<font color="Maroon" title="c13">D</font>,<font color="Maroon" title="c18">i</font>)</span>)</span>,<font color="Maroon" title="c12">rho</font>)</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="integr22.html#D1" onmouseover="rs('integr22/D1')" onmouseout="rh()">INTEGR22:def 1</a></span>
;<br/><span class="lab"><font color="Green" title="E57">A242</font></span>:  <a href="dualsp05.html#K3" title="DUALSP05:func.3">Dp2</a> (<font color="Maroon" title="c12">rho</font>,<font color="Maroon" title="c13">D</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c18">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>) = 
<font color="Maroon" title="c12">rho</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">D</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c18">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:22_2_4_1_2"><span class="lab"><font color="Green" title="E54">A238</font></span></a>, <a class="ref" href="dualsp05.html#D3" target="_self" onmouseover="rs('dualsp05/D3')" onmouseout="rh()">defDp2</a></span>
<br/>.= 
<font color="Maroon" title="c12">rho</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">D</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">i</font></span>)</span>

;<br/> <a href="absvalue.html#K1" title="ABSVALUE:func.1">sgn</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="dualsp05.html#K3" title="DUALSP05:func.3">Dp2</a> (<font color="Maroon" title="c12">rho</font>,<font color="Maroon" title="c13">D</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c18">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>)</span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p2">(<span class="default"><a href="dualsp05.html#K3" title="DUALSP05:func.3">Dp2</a> (<font color="Maroon" title="c12">rho</font>,<font color="Maroon" title="c13">D</font>,<font color="Maroon" title="c18">i</font>)</span>)</span></span>)</span> = 
 <a href="absvalue.html#K1" title="ABSVALUE:func.1">sgn</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c12">rho</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><a href="seq_4.html#K2" title="SEQ_4:func.2">upper_bound</a> <span class="p4">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c13">D</font>,<font color="Maroon" title="c18">i</font>)</span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c12">rho</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><a href="seq_4.html#K3" title="SEQ_4:func.3">lower_bound</a> <span class="p4">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c13">D</font>,<font color="Maroon" title="c18">i</font>)</span>)</span></span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:22_2_4_1_2"><span class="lab"><font color="Green" title="E53">A237</font></span></a>, <a class="txt" href="#E10:22_2_4_1_2"><span class="lab"><font color="Green" title="E57">A242</font></span></a>, <a class="txt" href="#E1:22_2_4_1_2"><span class="lab"><font color="Green" title="E50">A234</font></span></a>, <a class="ref" href="dualsp05.html#D3" target="_self" onmouseover="rs('dualsp05/D3')" onmouseout="rh()">defDp2</a></span>
<br/>.= 
 <a href="absvalue.html#K1" title="ABSVALUE:func.1">sgn</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> (<font color="Maroon" title="c13">D</font>,<font color="Maroon" title="c18">i</font>)</span>)</span>,<font color="Maroon" title="c12">rho</font>)</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="integr22.html#D1" onmouseover="rs('integr22/D1')" onmouseout="rh()">INTEGR22:def 1</a></span>
;<br/><span class="kw">hence </span><a NAME="E12:22_2_4_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c7">f</font> <a href="finseqop.html#K4" title="FINSEQOP:func.4">*</a> <font color="Maroon" title="c15">s</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">i</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"><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> (<font color="Maroon" title="c13">D</font>,<font color="Maroon" title="c18">i</font>)</span>)</span>,<font color="Maroon" title="c12">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="txt" href="#E9:22_2_4_1_2"><span class="lab"><font color="Green" title="E56">A241</font></span></a>, <a class="ref" href="dualsp05.html#T2" target="_self" onmouseover="rs('dualsp05/T2')" onmouseout="rh()">LM86</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></div><span class="kw">end;</span></div>

</div><span class="kw">end;</span></div>
<a NAME="E20:22_2"/>
 for <font color="Olive" title="b1">i</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Olive" title="b1">i</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> <font color="Maroon" title="c14">K</font> holds <br/><span class="p1">(<span class="default"><font color="Maroon" title="c7">f</font> <a href="finseqop.html#K4" title="FINSEQOP:func.4">*</a> <font color="Maroon" title="c15">s</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">K</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="22_2_5"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c18">i</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="c18">i</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> <font color="Maroon" title="c14">K</font> implies <span class="p1">(<span class="default"><font color="Maroon" title="c7">f</font> <a href="finseqop.html#K4" title="FINSEQOP:func.4">*</a> <font color="Maroon" title="c15">s</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">K</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">i</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:22_2_5"/>
<font color="Maroon" title="c18">i</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> <font color="Maroon" title="c14">K</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">(<span class="default"><font color="Maroon" title="c7">f</font> <a href="finseqop.html#K4" title="FINSEQOP:func.4">*</a> <font color="Maroon" title="c15">s</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">K</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">i</font></span><br/>

<a NAME="E2:22_2_5"/><span class="kw">then </span>
<font color="Maroon" title="c18">i</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> <font color="Maroon" title="c13">D</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:22_2"><span class="lab"><font color="Green" title="E38">A215</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="E3:22_2_5"/><span class="kw">then </span><span class="lab"><font color="Green" title="E41">A244</font></span>: 
<font color="Maroon" title="c18">i</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> <font color="Maroon" title="c13">D</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="E4:22_2_5"/><span class="kw">then </span>
<font color="Maroon" title="c14">K</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">i</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"><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> (<font color="Maroon" title="c13">D</font>,<font color="Maroon" title="c18">i</font>)</span>)</span>,<font color="Maroon" title="c12">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="integr22.html#D2" onmouseover="rs('integr22/D2')" onmouseout="rh()">INTEGR22:def 2</a></span>;<br/>
<span class="kw">hence </span><a NAME="E5:22_2_5"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c7">f</font> <a href="finseqop.html#K4" title="FINSEQOP:func.4">*</a> <font color="Maroon" title="c15">s</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">K</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">i</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:22_2"><span class="lab"><font color="Green" title="E40">A217</font></span></a>, <a class="txt" href="#E3:22_2_5"><span class="lab"><font color="Green" title="E41">A244</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="E21:22_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E41">A245</font></span>: 
<font color="Maroon" title="c7">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K4" title="RLVECT_1:func.4">Sum</a> <font color="Maroon" title="c15">s</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <font color="Maroon" title="c14">K</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:22_2"><span class="lab"><font color="Green" title="E35">A14</font></span></a>, <a class="txt" href="#E18:22_2"><span class="lab"><font color="Green" title="E39">A216</font></span></a>, <a class="ref" href="finseq_1.html#T13" onmouseover="rs('finseq_1/T13')" onmouseout="rh()">FINSEQ_1:13</a></span>;<br/>
<a NAME="E22:22_2"/><span class="lab"><font color="Green" title="E42">A246</font></span>: 
<font color="Maroon" title="c7">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K4" title="RLVECT_1:func.4">Sum</a> <font color="Maroon" title="c15">s</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</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="c7">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p3">(<span class="default"><a href="rlvect_1.html#K4" title="RLVECT_1:func.4">Sum</a> <font color="Maroon" title="c15">s</font></span>)</span></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="absvalue.html#T4" onmouseover="rs('absvalue/T4')" onmouseout="rh()">ABSVALUE:4</a></span>;<br/>
<a NAME="E23:22_2"/><span class="lab"><font color="Green" title="E43">A247</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="c7">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p3">(<span class="default"><a href="rlvect_1.html#K4" title="RLVECT_1:func.4">Sum</a> <font color="Maroon" title="c15">s</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="c8">F</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"><a href="normsp_0.html#K2" title="NORMSP_0:func.2">||.</a><span class="default"><span class="p2">(<span class="default"><a href="rlvect_1.html#K4" title="RLVECT_1:func.4">Sum</a> <font color="Maroon" title="c15">s</font></span>)</span></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="#E9:22"><span class="lab"><font color="Green" title="E27">A6</font></span></a>, <a class="ref" href="dualsp01.html#T26" onmouseover="rs('dualsp01/T26')" onmouseout="rh()">DUALSP01:26</a></span>;<br/>
<a NAME="E24:22_2"/>
<span class="p1"><a href="normsp_0.html#K2" title="NORMSP_0:func.2">||.</a><span class="default"><font color="Maroon" title="c8">F</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"><a href="normsp_0.html#K2" title="NORMSP_0:func.2">||.</a><span class="default"><span class="p2">(<span class="default"><a href="rlvect_1.html#K4" title="RLVECT_1:func.4">Sum</a> <font color="Maroon" title="c15">s</font></span>)</span></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> <span class="p1"><a href="normsp_0.html#K2" title="NORMSP_0:func.2">||.</a><span class="default"><font color="Maroon" title="c8">F</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> 1
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:22_2"><span class="lab"><font color="Green" title="E37">A214</font></span></a>, <a class="ref" href="xreal_1.html#T64" onmouseover="rs('xreal_1/T64')" onmouseout="rh()">XREAL_1:64</a></span>;<br/>
<a NAME="E25:22_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="c7">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p3">(<span class="default"><a href="rlvect_1.html#K4" title="RLVECT_1:func.4">Sum</a> <font color="Maroon" title="c15">s</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="c8">F</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="#E23:22_2"><span class="lab"><font color="Green" title="E43">A247</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="E26:22_2"/>
 <a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <font color="Maroon" title="c14">K</font> <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="c2">x</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="#E9:22"><span class="lab"><font color="Green" title="E27">A6</font></span></a>, <a class="txt" href="#E21:22_2"><span class="lab"><font color="Green" title="E41">A245</font></span></a>, <a class="txt" href="#E22:22_2"><span class="lab"><font color="Green" title="E42">A246</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>
<span class="kw">reconsider </span><font color="Maroon" title="c13">d</font> = <span class="p1"><a href="normsp_0.html#K2" title="NORMSP_0:func.2">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="normsp_0.html#K2" title="NORMSP_0:func.2">.||</a></span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1 as   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a> ;<br/>
<a NAME="E18:22"/>
<font color="Maroon" title="c1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><span class="p2">(<span class="default"><a href="seq_4.html#K3" title="SEQ_4:func.3">lower_bound</a> <font color="Maroon" title="c1">A</font></span>)</span>,<span class="p2">(<span class="default"><a href="seq_4.html#K2" title="SEQ_4:func.2">upper_bound</a> <font color="Maroon" title="c1">A</font></span>)</span></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="integra1.html#T4" onmouseover="rs('integra1/T4')" onmouseout="rh()">INTEGRA1:4</a></span>;<br/>
<a NAME="E19:22"/><span class="kw">then </span><span class="lab"><font color="Green" title="E31">A249</font></span>: 
 <a href="seq_4.html#K3" title="SEQ_4:func.3">lower_bound</a> <font color="Maroon" title="c1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">a</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:22"><span class="lab"><font color="Green" title="E28">A7</font></span></a>, <a class="ref" href="integra1.html#T5" onmouseover="rs('integra1/T5')" onmouseout="rh()">INTEGRA1:5</a></span>;<br/>
<a NAME="E20:22"/><span class="kw">then </span>
 for <font color="Olive" title="b1">D</font> being    <a href="integra1.html#M1" title="INTEGRA1:mode.1">Division</a> of <font color="Maroon" title="c1">A</font><br/>  for <font color="Olive" title="b2">K</font> being    <a href="integr22.html#M1" title="INTEGR22:mode.1">var_volume</a> of <font color="Maroon" title="c12">rho</font>,<font color="Olive" title="b1">D</font> holds   <a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <font color="Olive" title="b2">K</font> <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="c2">x</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="#E1:22"><span class="lab"><font color="Green" title="E22">A1</font></span></a>, <a class="txt" href="#E16:22"><span class="lab"><font color="Green" title="E30">A9</font></span></a>, <a class="ref" href="dualsp05.html#T14" target="_self" onmouseover="rs('dualsp05/T14')" onmouseout="rh()">LM95</a></span>;<br/>
<a NAME="E21:22"/><span class="kw">then </span><span class="lab"><font color="Green" title="E32">B250</font></span>: 
 for <font color="Olive" title="b1">t</font> being    <a href="integra1.html#M1" title="INTEGRA1:mode.1">Division</a> of <font color="Maroon" title="c1">A</font><br/>  for <font color="Olive" title="b2">F0</font> being    <a href="integr22.html#M1" title="INTEGR22:mode.1">var_volume</a> of <font color="Maroon" title="c12">rho</font>,<font color="Olive" title="b1">t</font> holds   <a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <font color="Olive" title="b2">F0</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c13">d</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T39" onmouseover="rs('xreal_1/T39')" onmouseout="rh()">XREAL_1:39</a></span>;<br/>
<a NAME="E22:22"/><span class="kw">then </span><span class="lab"><font color="Green" title="E33">A251</font></span>: 
<font color="Maroon" title="c12">rho</font> is  <a href="integr22.html#V1" title="INTEGR22:attr.1">bounded_variation</a> 
 
;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c14">VD</font> being   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</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><span class="kw"> such that </span><br/><a NAME="E24:22"/>
<font color="Maroon" title="c14">VD</font> is  <a href="xxreal_2.html#V4" title="XXREAL_2:attr.4">bounded_above</a> 
 <span class="kw">and </span><br/><a NAME="E25:22"/><span class="lab"><font color="Green" title="E34">A253</font></span>: 
<font color="Maroon" title="c14">VD</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">r</font> where <font color="Olive" title="b1">r</font> is   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a> :  ex <font color="Olive" title="b2">t</font> being    <a href="integra1.html#M1" title="INTEGRA1:mode.1">Division</a> of <font color="Maroon" title="c1">A</font> ex <font color="Olive" title="b3">F0</font> being    <a href="integr22.html#M1" title="INTEGR22:mode.1">var_volume</a> of <font color="Maroon" title="c12">rho</font>,<font color="Olive" title="b2">t</font> st <font color="Olive" title="b1">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <font color="Olive" title="b3">F0</font> </span> } </span> 
 <span class="kw">and </span><br/><a NAME="E26:22"/><span class="lab"><font color="Green" title="E35">A254</font></span>: 
 <a href="integr22.html#K2" title="INTEGR22:func.2">total_vd</a> <font color="Maroon" title="c12">rho</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="seq_4.html#K2" title="SEQ_4:func.2">upper_bound</a> <font color="Maroon" title="c14">VD</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="integr22.html#D4" onmouseover="rs('integr22/D4')" onmouseout="rh()">INTEGR22:def 4</a></span>;<br/>
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="22_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> <font color="Maroon" title="c14">VD</font> holds <br/><font color="Olive" title="b1">k</font> <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="c2">x</font></span><a href="normsp_0.html#K2" title="NORMSP_0:func.2">.||</a></span></span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c15">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="c15">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c14">VD</font> implies <font color="Maroon" title="c15">k</font> <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="c2">x</font></span><a href="normsp_0.html#K2" title="NORMSP_0:func.2">.||</a></span> )</span><br/><span class="kw">assume </span><a NAME="E1:22_3"/>
<font color="Maroon" title="c15">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c14">VD</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="c15">k</font> <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="c2">x</font></span><a href="normsp_0.html#K2" title="NORMSP_0:func.2">.||</a></span></span><br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c16">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="E3:22_3"/><span class="lab"><font color="Green" title="E36">A255</font></span>: 
<font color="Maroon" title="c15">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">r</font>
 <span class="kw">and </span><br/><a NAME="E4:22_3"/><span class="lab"><font color="Green" title="E37">A256</font></span>: 
 ex <font color="Olive" title="b1">t</font> being    <a href="integra1.html#M1" title="INTEGRA1:mode.1">Division</a> of <font color="Maroon" title="c1">A</font> ex <font color="Olive" title="b2">F0</font> being    <a href="integr22.html#M1" title="INTEGR22:mode.1">var_volume</a> of <font color="Maroon" title="c12">rho</font>,<font color="Olive" title="b1">t</font> st <font color="Maroon" title="c16">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <font color="Olive" title="b2">F0</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E25:22"><span class="lab"><font color="Green" title="E34">A253</font></span></a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c17">t</font> being    <a href="integra1.html#M1" title="INTEGRA1:mode.1">Division</a> of <font color="Maroon" title="c1">A</font>, <font color="Maroon" title="c18">F0</font> being    <a href="integr22.html#M1" title="INTEGR22:mode.1">var_volume</a> of <font color="Maroon" title="c12">rho</font>,<font color="Maroon" title="c17">t</font><span class="kw"> such that </span><br/><a NAME="E6:22_3"/><span class="lab"><font color="Green" title="E38">A257</font></span>: 
<font color="Maroon" title="c16">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <font color="Maroon" title="c18">F0</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:22_3"><span class="lab"><font color="Green" title="E37">A256</font></span></a></span>;<br/><span class="kw">thus </span><a NAME="E7:22_3"/>
<font color="Maroon" title="c15">k</font> <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="c2">x</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="#E19:22"><span class="lab"><font color="Green" title="E31">A249</font></span></a>, <a class="txt" href="#E1:22"><span class="lab"><font color="Green" title="E22">A1</font></span></a>, <a class="txt" href="#E16:22"><span class="lab"><font color="Green" title="E30">A9</font></span></a>, <a class="ref" href="dualsp05.html#T14" target="_self" onmouseover="rs('dualsp05/T14')" onmouseout="rh()">LM95</a>, <a class="txt" href="#E3:22_3"><span class="lab"><font color="Green" title="E36">A255</font></span></a>, <a class="txt" href="#E6:22_3"><span class="lab"><font color="Green" title="E38">A257</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="E28:22"/><span class="kw">then </span><span class="lab"><font color="Green" title="E36">A258</font></span>: 
 <a href="integr22.html#K2" title="INTEGR22:func.2">total_vd</a> <font color="Maroon" title="c12">rho</font> <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="c2">x</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="#E26:22"><span class="lab"><font color="Green" title="E35">A254</font></span></a>, <a class="ref" href="seq_4.html#T45" onmouseover="rs('seq_4/T45')" onmouseout="rh()">SEQ_4:45</a></span>;<br/>
<a NAME="E29:22"/><span class="lab"><font color="Green" title="E37">A259</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="c2">x</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="c12">rho</font>)
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="22_4"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c15">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"> (  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c15">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">A</font> implies <font color="Maroon" title="c2">x</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">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="c15">u</font>,<font color="Maroon" title="c12">rho</font>) )</span><br/>

<span class="kw">assume </span><a NAME="E1:22_4"/><span class="lab"><font color="Green" title="E38">A260</font></span>: 
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c15">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">A</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="c2">x</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">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="c15">u</font>,<font color="Maroon" title="c12">rho</font>)</span><br/>

<a NAME="E2:22_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E39">A261</font></span>: 
 for <font color="Olive" title="b1">r</font> being   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a>  st  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">r</font> holds <br/> ex <font color="Olive" title="b2">s</font> being   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a> st <br/>(  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b2">s</font> &amp; (  for <font color="Olive" title="b3">x1</font>, <font color="Olive" title="b4">x2</font> being   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a>  st <font color="Olive" title="b3">x1</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="c15">u</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c1">A</font></span>)</span> &amp; <font color="Olive" title="b4">x2</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="c15">u</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c1">A</font></span>)</span> &amp; <span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b3">x1</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Olive" title="b4">x2</font></span>)</span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b2">s</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"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c15">u</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b3">x1</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c15">u</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b4">x2</font></span>)</span></span>)</span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">r</font> ) )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="fcont_2.html#D1" onmouseover="rs('fcont_2/D1')" onmouseout="rh()">FCONT_2:def 1</a>, <a class="ref" href="fcont_2.html#T11" onmouseover="rs('fcont_2/T11')" onmouseout="rh()">FCONT_2:11</a></span>;<br/>
<a NAME="E3:22_4"/><span class="lab"><font color="Green" title="E40">B262</font></span>: 
<font color="Maroon" title="c15">u</font> <a href="integr22.html#R1" title="INTEGR22:pred.1">is_RiemannStieltjes_integrable_with</a> <font color="Maroon" title="c12">rho</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E22:22"><span class="lab"><font color="Green" title="E33">A251</font></span></a>, <a class="txt" href="#E1:22_4"><span class="lab"><font color="Green" title="E38">A260</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="c16">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="E5:22_4"/><span class="lab"><font color="Green" title="E41">A263</font></span>: 
(  <a href="integra3.html#K2" title="INTEGRA3:func.2">delta</a> <font color="Maroon" title="c16">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="c16">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>  &amp; (  for <font color="Olive" title="b1">n</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>   ex <font color="Olive" title="b2">Tn</font> being    <a href="integra1.html#M1" title="INTEGRA1:mode.1">Division</a> of <font color="Maroon" title="c1">A</font> st <br/>( <font color="Olive" title="b2">Tn</font> <a href="integra4.html#R1" title="INTEGRA4:pred.1">divide_into_equal</a> <font color="Olive" title="b1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <font color="Maroon" title="c16">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">Tn</font> ) ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:22"><span class="lab"><font color="Green" title="E22">A1</font></span></a>, <a class="ref" href="integra6.html#T16" onmouseover="rs('integra6/T16')" onmouseout="rh()">INTEGRA6:16</a></span>;<br/>
<a NAME="E6:22_4"/><span class="lab"><font color="Green" title="E42">A264</font></span>: 
 for <font color="Olive" title="b1">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  holds <font color="Maroon" title="c9">a</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Olive" title="b1">n</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="22_4_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c17">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"> <font color="Maroon" title="c9">a</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c17">n</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1</span><br/>

<a NAME="E1:22_4_1"/>
<font color="Maroon" title="c17">n</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal1.html#D12" onmouseover="rs('ordinal1/D12')" onmouseout="rh()">ORDINAL1:def 12</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c18">Tn</font> being    <a href="integra1.html#M1" title="INTEGRA1:mode.1">Division</a> of <font color="Maroon" title="c1">A</font><span class="kw"> such that </span><br/><a NAME="E3:22_4_1"/><span class="lab"><font color="Green" title="E43">A265</font></span>: 
( <font color="Maroon" title="c18">Tn</font> <a href="integra4.html#R1" title="INTEGRA4:pred.1">divide_into_equal</a> <font color="Maroon" title="c17">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <font color="Maroon" title="c16">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c17">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">Tn</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:22_4"><span class="lab"><font color="Green" title="E41">A263</font></span></a></span>;<br/>
<a NAME="E4:22_4_1"/><span class="lab"><font color="Green" title="E44">A266</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c18">Tn</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:22_4_1"><span class="lab"><font color="Green" title="E43">A265</font></span></a>, <a class="ref" href="integra4.html#D1" onmouseover="rs('integra4/D1')" onmouseout="rh()">INTEGRA4:def 1</a></span>;<br/>
<a NAME="E5:22_4_1"/>
<font color="Maroon" title="c17">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <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"><font color="Maroon" title="c17">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T4" onmouseover="rs('finseq_1/T4')" onmouseout="rh()">FINSEQ_1:4</a></span>;<br/>
<a NAME="E6:22_4_1"/><span class="kw">then </span>
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c17">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <font color="Maroon" title="c17">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c18">Tn</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T1" onmouseover="rs('finseq_1/T1')" onmouseout="rh()">FINSEQ_1:1</a>, <a class="txt" href="#E3:22_4_1"><span class="lab"><font color="Green" title="E43">A265</font></span></a>, <a class="ref" href="integra4.html#D1" onmouseover="rs('integra4/D1')" onmouseout="rh()">INTEGRA4:def 1</a></span>;<br/>
<a NAME="E7:22_4_1"/><span class="kw">then </span>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c18">Tn</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_0.html#T2" onmouseover="rs('xxreal_0/T2')" onmouseout="rh()">XXREAL_0:2</a></span>;<br/>
<a NAME="E8:22_4_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E45">B268</font></span>: 
1 <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c18">Tn</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_3.html#T25" onmouseover="rs('finseq_3/T25')" onmouseout="rh()">FINSEQ_3:25</a></span>;<br/>
<a NAME="E9:22_4_1"/>
 <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><a href="integra1.html#K3" title="INTEGRA1:func.3">vol</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c18">Tn</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:22"><span class="lab"><font color="Green" title="E22">A1</font></span></a>, <a class="txt" href="#E4:22_4_1"><span class="lab"><font color="Green" title="E44">A266</font></span></a>, <a class="ref" href="xreal_1.html#T139" onmouseover="rs('xreal_1/T139')" onmouseout="rh()">XREAL_1:139</a></span>;<br/>
<a NAME="E10:22_4_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="seq_4.html#K3" title="SEQ_4:func.3">lower_bound</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><a href="seq_4.html#K3" title="SEQ_4:func.3">lower_bound</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="integra1.html#K3" title="INTEGRA1:func.3">vol</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c18">Tn</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> 1</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T8" onmouseover="rs('xreal_1/T8')" onmouseout="rh()">XREAL_1:8</a></span>;<br/>
<span class="kw">hence </span><a NAME="E11:22_4_1"/>
<font color="Maroon" title="c9">a</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c17">n</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:22_4_1"><span class="lab"><font color="Green" title="E43">A265</font></span></a>, <a class="txt" href="#E19:22"><span class="lab"><font color="Green" title="E31">A249</font></span></a>, <a class="txt" href="#E8:22_4_1"><span class="lab"><font color="Green" title="E45">B268</font></span></a>, <a class="ref" href="integra4.html#D1" onmouseover="rs('integra4/D1')" onmouseout="rh()">INTEGRA4:def 1</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">set </span><font color="Maroon" title="c17">S</font> =  the    <a href="integr22.html#M3" title="INTEGR22:mode.3">middle_volume_Sequence</a> of <font color="Maroon" title="c12">rho</font>,<font color="Maroon" title="c15">u</font>,<font color="Maroon" title="c16">T</font>;<br/>
<a NAME="E7:22_4"/><span class="lab"><font color="Green" title="E43">A269</font></span>: 
<font color="Maroon" title="c15">u</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="c0sp2.html#K5" title="C0SP2:func.5">R_Normed_Algebra_of_ContinuousFunctions</a> <span class="p2">(<span class="default"><a href="dualsp05.html#K1" title="DUALSP05:func.1">ClstoCmp</a> <font color="Maroon" title="c1">A</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:22_4"><span class="lab"><font color="Green" title="E38">A260</font></span></a>, <a class="ref" href="dualsp05.html#T6" target="_self" onmouseover="rs('dualsp05/T6')" onmouseout="rh()">Th80</a></span>;<br/>
<span class="kw">defpred </span><font color="Maroon">S<sub>1</sub></font>[   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a> ,   <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a> ] <span class="kw">means </span> ex <font color="Olive" title="b1">p</font> being    <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>  st <br/>( <font color="Olive" title="b1">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> $2 &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> $1</span>)</span> &amp; (  for <font color="Olive" title="b2">i</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Olive" title="b2">i</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="c16">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> $1</span>)</span> holds <br/>( <font color="Olive" title="b1">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">i</font> <a href="tarski.html#R2" title="TARSKI: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="c15">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="c16">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> $1</span>)</span>,<font color="Olive" title="b2">i</font>)</span>)</span></span>)</span> &amp;  ex <font color="Olive" title="b3">z</font> being   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a> st <br/>( <font color="Olive" title="b3">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">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="c16">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> $1</span>)</span>,<font color="Olive" title="b2">i</font>)</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">i</font></span>)</span> &amp; <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="c12">rho</font>,<font color="Maroon" title="c15">u</font>,<font color="Maroon" title="c16">T</font> <a href="integr22.html#K3" title="INTEGR22:func.3">.</a> $1</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">z</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="c16">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> $1</span>)</span>,<font color="Olive" title="b2">i</font>)</span>)</span>,<font color="Maroon" title="c12">rho</font>)</span>)</span> ) ) ) );<br/>
<a NAME="E8:22_4"/><span class="lab"><font color="Green" title="E44">A270</font></span>: 
 for <font color="Olive" title="b1">k</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>   ex <font color="Olive" title="b2">p</font> being    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">k</font>,<font color="Olive" title="b2">p</font>]
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="22_4_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c18">k</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">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">  ex <font color="Olive" title="b1">p</font> being    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c18">k</font>,<font color="Olive" title="b1">p</font>]</span><br/>

<span class="kw">defpred </span><font color="Maroon">S<sub>2</sub></font>[  <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>,   <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a> ] <span class="kw">means </span>( $2 <a href="tarski.html#R2" title="TARSKI: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="c15">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="c16">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c18">k</font></span>)</span>,$1)</span>)</span></span>)</span> &amp;  ex <font color="Olive" title="b1">c</font> being   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a> st <br/>( <font color="Olive" title="b1">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">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="c16">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c18">k</font></span>)</span>,$1)</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> $2 &amp; <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="c12">rho</font>,<font color="Maroon" title="c15">u</font>,<font color="Maroon" title="c16">T</font> <a href="integr22.html#K3" title="INTEGR22:func.3">.</a> <font color="Maroon" title="c18">k</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> $1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">c</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="c16">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c18">k</font></span>)</span>,$1)</span>)</span>,<font color="Maroon" title="c12">rho</font>)</span>)</span> ) );<br/>
<a NAME="E1:22_4_2"/><span class="lab"><font color="Green" title="E45">A271</font></span>: 
 <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="c16">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c18">k</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c18">k</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="E2:22_4_2"/><span class="lab"><font color="Green" title="E46">A272</font></span>: 
 for <font color="Olive" title="b1">i</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Olive" title="b1">i</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="c16">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c18">k</font></span>)</span></span>)</span> holds <br/> ex <font color="Olive" title="b2">x</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  st <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">x</font>]
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="22_4_2_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c19">i</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="c19">i</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="c16">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c18">k</font></span>)</span></span>)</span> implies  ex <font color="Olive" title="b1">x</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  st <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c19">i</font>,<font color="Olive" title="b1">x</font>] )</span><br/>

<span class="kw">assume </span><a NAME="E1:22_4_2_1"/>
<font color="Maroon" title="c19">i</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="c16">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c18">k</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">  ex <font color="Olive" title="b1">x</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  st <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c19">i</font>,<font color="Olive" title="b1">x</font>]</span><br/>

<a NAME="E2:22_4_2_1"/><span class="kw">then </span>
<font color="Maroon" title="c19">i</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="c16">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c18">k</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">then </span><span class="kw">consider </span><font color="Maroon" title="c20">c</font> being   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a><span class="kw"> such that </span><br/><a NAME="E4:22_4_2_1"/><span class="lab"><font color="Green" title="E47">A273</font></span>: 
( <font color="Maroon" title="c20">c</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="c15">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="c16">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c18">k</font></span>)</span>,<font color="Maroon" title="c19">i</font>)</span>)</span></span>)</span> &amp; <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="c12">rho</font>,<font color="Maroon" title="c15">u</font>,<font color="Maroon" title="c16">T</font> <a href="integr22.html#K3" title="INTEGR22:func.3">.</a> <font color="Maroon" title="c18">k</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">c</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="c16">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c18">k</font></span>)</span>,<font color="Maroon" title="c19">i</font>)</span>)</span>,<font color="Maroon" title="c12">rho</font>)</span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E22:22"><span class="lab"><font color="Green" title="E33">A251</font></span></a>, <a class="txt" href="#E1:22_4"><span class="lab"><font color="Green" title="E38">A260</font></span></a>, <a class="ref" href="integr22.html#D5" onmouseover="rs('integr22/D5')" onmouseout="rh()">INTEGR22:def 5</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c21">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> <span class="kw"> such that </span><br/><a NAME="E6:22_4_2_1"/><span class="lab"><font color="Green" title="E48">A274</font></span>: 
( <font color="Maroon" title="c21">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="c15">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="c16">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c18">k</font></span>)</span>,<font color="Maroon" title="c19">i</font>)</span>)</span></span>)</span> &amp; <font color="Maroon" title="c20">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">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="c16">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c18">k</font></span>)</span>,<font color="Maroon" title="c19">i</font>)</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">x</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:22_4_2_1"><span class="lab"><font color="Green" title="E47">A273</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">reconsider </span><font color="Maroon" title="c22">x</font> = <font color="Maroon" title="c21">x</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:22_4_2_1"><span class="lab"><font color="Green" title="E48">A274</font></span></a></span>;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c22">x</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">S<sub>2</sub></font>[<font color="Maroon" title="c19">i</font>,<font color="Maroon" title="c22">x</font>]</span><br/>

<span class="kw">thus </span><a NAME="E8:22_4_2_1"/>
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c19">i</font>,<font color="Maroon" title="c22">x</font>]
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:22_4_2_1"><span class="lab"><font color="Green" title="E47">A273</font></span></a>, <a class="txt" href="#E6:22_4_2_1"><span class="lab"><font color="Green" title="E48">A274</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">consider </span><font color="Maroon" title="c19">p</font> being    <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> <span class="kw"> such that </span><br/><a NAME="E4:22_4_2"/><span class="lab"><font color="Green" title="E47">A275</font></span>: 
(  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c19">p</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> <span class="p2">(<span class="default"><font color="Maroon" title="c16">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c18">k</font></span>)</span></span>)</span> &amp; (  for <font color="Olive" title="b1">i</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Olive" title="b1">i</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="c16">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c18">k</font></span>)</span></span>)</span> holds <br/><font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">i</font>,<font color="Maroon" title="c19">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font>] ) )
 <span class="kw">from</span> <span class="lab"><a class="ref" href="finseq_1.html#S5" onmouseover="rs('finseq_1/S5')" onmouseout="rh()">FINSEQ_1:sch 5</a>(<a class="txt" href="#E2:22_4_2"><span class="lab"><font color="Green" title="E46">A272</font></span></a>);<br/></span>
<span class="kw">take </span>
<font color="Maroon" title="c19">p</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="c19">p</font> is    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a>  &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c18">k</font>,<font color="Maroon" title="c19">p</font>] )</span><br/>

<a NAME="E5:22_4_2"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c19">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c18">k</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:22_4_2"><span class="lab"><font color="Green" title="E47">A275</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/>
<span class="kw">hence </span><a NAME="E6:22_4_2"/>
( <font color="Maroon" title="c19">p</font> is    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a>  &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c18">k</font>,<font color="Maroon" title="c19">p</font>] )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:22_4_2"><span class="lab"><font color="Green" title="E45">A271</font></span></a>, <a class="txt" href="#E4:22_4_2"><span class="lab"><font color="Green" title="E47">A275</font></span></a>, <a class="ref" href="finseq_1.html#D11" onmouseover="rs('finseq_1/D11')" onmouseout="rh()">FINSEQ_1:def 11</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">consider </span><font color="Maroon" title="c18">F</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">sequence</a> of <span class="p1">(<span class="default"><a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a></span>)</span><span class="kw"> such that </span><br/><a NAME="E10:22_4"/><span class="lab"><font color="Green" title="E45">A276</font></span>: 
 for <font color="Olive" title="b1">x</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">x</font>,<font color="Maroon" title="c18">F</font> <a href="finseq_2.html#K8" title="FINSEQ_2:func.8">.</a> <font color="Olive" title="b1">x</font>]
 <span class="kw">from</span> <span class="lab"><a class="ref" href="funct_2.html#S3" onmouseover="rs('funct_2/S3')" onmouseout="rh()">FUNCT_2:sch 3</a>(<a class="txt" href="#E8:22_4"><span class="lab"><font color="Green" title="E44">A270</font></span></a>);<br/></span>
<span class="kw">defpred </span><font color="Maroon">S<sub>2</sub></font>[   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a> ,   <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> ] <span class="kw">means </span> ex <font color="Olive" title="b1">q</font> being   <a href="struct_0.html#NM7" title="STRUCT_0:NM.7">FinSequence</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> st <br/>(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> $1</span>)</span> &amp; $2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">Sum</a> <font color="Olive" title="b1">q</font> &amp; (  for <font color="Olive" title="b2">i</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Olive" title="b2">i</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="c16">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> $1</span>)</span> holds <br/> ex <font color="Olive" title="b3">r</font> being   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a> st <br/>( <span class="p1">(<span class="default"><font color="Maroon" title="c18">F</font> <a href="finseq_2.html#K8" title="FINSEQ_2:func.8">.</a> $1</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">i</font> <a href="tarski.html#R2" title="TARSKI: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="c15">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="c16">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> $1</span>)</span>,<font color="Olive" title="b2">i</font>)</span>)</span></span>)</span> &amp; <font color="Olive" title="b3">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">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="c16">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> $1</span>)</span>,<font color="Olive" title="b2">i</font>)</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c18">F</font> <a href="finseq_2.html#K8" title="FINSEQ_2:func.8">.</a> $1</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">i</font></span>)</span> &amp; <font color="Olive" title="b1">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">r</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="dualsp05.html#K2" title="DUALSP05:func.2">Dp1</a> (<font color="Maroon" title="c11">m</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c16">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> $1</span>)</span>,<span class="p3">(<span class="default"><font color="Olive" title="b2">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>)</span>)</span> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <span class="p2">(<span class="default"><a href="dualsp05.html#K2" title="DUALSP05:func.2">Dp1</a> (<font color="Maroon" title="c11">m</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c16">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> $1</span>)</span>,<font color="Olive" title="b2">i</font>)</span>)</span></span>)</span> ) ) );<br/>
<a NAME="E11:22_4"/><span class="lab"><font color="Green" title="E46">A277</font></span>: 
 for <font color="Olive" title="b1">k</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>   ex <font color="Olive" title="b2">x</font> being    <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="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> st <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">k</font>,<font color="Olive" title="b2">x</font>]
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="22_4_3"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c19">k</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">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">  ex <font color="Olive" title="b1">x</font> being    <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="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> st <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c19">k</font>,<font color="Olive" title="b1">x</font>]</span><br/>

<span class="kw">defpred </span><font color="Maroon">S<sub>3</sub></font>[  <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>,   <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> ] <span class="kw">means </span> ex <font color="Olive" title="b1">r</font> being   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a> st <br/>( <span class="p1">(<span class="default"><font color="Maroon" title="c18">F</font> <a href="finseq_2.html#K8" title="FINSEQ_2:func.8">.</a> <font color="Maroon" title="c19">k</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> $1 <a href="tarski.html#R2" title="TARSKI: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="c15">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="c16">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c19">k</font></span>)</span>,$1)</span>)</span></span>)</span> &amp; <font color="Olive" title="b1">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">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="c16">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c19">k</font></span>)</span>,$1)</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c18">F</font> <a href="finseq_2.html#K8" title="FINSEQ_2:func.8">.</a> <font color="Maroon" title="c19">k</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> $1</span>)</span> &amp; $2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">r</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="dualsp05.html#K2" title="DUALSP05:func.2">Dp1</a> (<font color="Maroon" title="c11">m</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c16">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c19">k</font></span>)</span>,<span class="p3">(<span class="default">$1 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>)</span>)</span> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <span class="p2">(<span class="default"><a href="dualsp05.html#K2" title="DUALSP05:func.2">Dp1</a> (<font color="Maroon" title="c11">m</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c16">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c19">k</font></span>)</span>,$1)</span>)</span></span>)</span> );<br/>
<a NAME="E1:22_4_3"/><span class="lab"><font color="Green" title="E47">A278</font></span>: 
 for <font color="Olive" title="b1">i</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Olive" title="b1">i</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="c16">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c19">k</font></span>)</span></span>)</span> holds <br/> ex <font color="Olive" title="b2">y</font> being    <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="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> st <font color="Maroon">S<sub>3</sub></font>[<font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">y</font>]
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="22_4_3_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c20">i</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="c20">i</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="c16">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c19">k</font></span>)</span></span>)</span> implies  ex <font color="Olive" title="b1">y</font> being    <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="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> st <font color="Maroon">S<sub>3</sub></font>[<font color="Maroon" title="c20">i</font>,<font color="Olive" title="b1">y</font>] )</span><br/>

<span class="kw">assume </span><a NAME="E1:22_4_3_1"/>
<font color="Maroon" title="c20">i</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="c16">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c19">k</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">  ex <font color="Olive" title="b1">y</font> being    <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="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> st <font color="Maroon">S<sub>3</sub></font>[<font color="Maroon" title="c20">i</font>,<font color="Olive" title="b1">y</font>]</span><br/>

<a NAME="E2:22_4_3_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E48">A279</font></span>: 
<font color="Maroon" title="c20">i</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="c16">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c19">k</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="c21">p</font> being    <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> <span class="kw"> such that </span><br/><a NAME="E4:22_4_3_1"/><span class="lab"><font color="Green" title="E49">A280</font></span>: 
( <font color="Maroon" title="c21">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">F</font> <a href="finseq_2.html#K8" title="FINSEQ_2:func.8">.</a> <font color="Maroon" title="c19">k</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c21">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c19">k</font></span>)</span> &amp; (  for <font color="Olive" title="b1">i</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Olive" title="b1">i</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="c16">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c19">k</font></span>)</span> holds <br/>( <font color="Maroon" title="c21">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font> <a href="tarski.html#R2" title="TARSKI: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="c15">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="c16">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c19">k</font></span>)</span>,<font color="Olive" title="b1">i</font>)</span>)</span></span>)</span> &amp;  ex <font color="Olive" title="b2">z</font> being   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a> st <br/>( <font color="Olive" title="b2">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">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="c16">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c19">k</font></span>)</span>,<font color="Olive" title="b1">i</font>)</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c21">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font></span>)</span> &amp; <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="c12">rho</font>,<font color="Maroon" title="c15">u</font>,<font color="Maroon" title="c16">T</font> <a href="integr22.html#K3" title="INTEGR22:func.3">.</a> <font color="Maroon" title="c19">k</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">z</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="c16">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c19">k</font></span>)</span>,<font color="Olive" title="b1">i</font>)</span>)</span>,<font color="Maroon" title="c12">rho</font>)</span>)</span> ) ) ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:22_4"><span class="lab"><font color="Green" title="E45">A276</font></span></a></span>;<br/>
<a NAME="E5:22_4_3_1"/>
<font color="Maroon" title="c21">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">i</font> <a href="tarski.html#R2" title="TARSKI: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="c15">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="c16">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c19">k</font></span>)</span>,<font color="Maroon" title="c20">i</font>)</span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:22_4_3_1"><span class="lab"><font color="Green" title="E48">A279</font></span></a>, <a class="txt" href="#E4:22_4_3_1"><span class="lab"><font color="Green" title="E49">A280</font></span></a></span>;<br/>
<a NAME="E6:22_4_3_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c15">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="c16">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c19">k</font></span>)</span>,<font color="Maroon" title="c20">i</font>)</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c21">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">i</font></span>)</span> <a href="tarski.html#R2" title="TARSKI: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="c15">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="c16">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c19">k</font></span>)</span>,<font color="Maroon" title="c20">i</font>)</span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#T3" onmouseover="rs('funct_1/T3')" onmouseout="rh()">FUNCT_1:3</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c22">r</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c15">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="c16">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c19">k</font></span>)</span>,<font color="Maroon" title="c20">i</font>)</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c21">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">i</font></span>)</span> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  ;<br/>
<a NAME="E8:22_4_3_1"/>
<font color="Maroon" title="c22">r</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="dualsp05.html#K2" title="DUALSP05:func.2">Dp1</a> (<font color="Maroon" title="c11">m</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c16">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c19">k</font></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon" title="c20">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>)</span>)</span> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <span class="p2">(<span class="default"><a href="dualsp05.html#K2" title="DUALSP05:func.2">Dp1</a> (<font color="Maroon" title="c11">m</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c16">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c19">k</font></span>)</span>,<font color="Maroon" title="c20">i</font>)</span>)</span></span>)</span> is    <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="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>
 
;<br/>
<span class="kw">hence </span><a NAME="E9:22_4_3_1"/>
 ex <font color="Olive" title="b1">y</font> being    <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="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> st <font color="Maroon">S<sub>3</sub></font>[<font color="Maroon" title="c20">i</font>,<font color="Olive" title="b1">y</font>]
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:22_4_3_1"><span class="lab"><font color="Green" title="E49">A280</font></span></a>, <a class="txt" href="#E2:22_4_3_1"><span class="lab"><font color="Green" title="E48">A279</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">consider </span><font color="Maroon" title="c20">q</font> being   <a href="struct_0.html#NM7" title="STRUCT_0:NM.7">FinSequence</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"> such that </span><br/><a NAME="E3:22_4_3"/><span class="lab"><font color="Green" title="E48">A282</font></span>: 
(  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c20">q</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> <span class="p2">(<span class="default"><font color="Maroon" title="c16">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c19">k</font></span>)</span></span>)</span> &amp; (  for <font color="Olive" title="b1">i</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Olive" title="b1">i</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="c16">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c19">k</font></span>)</span></span>)</span> holds <br/><font color="Maroon">S<sub>3</sub></font>[<font color="Olive" title="b1">i</font>,<font color="Maroon" title="c20">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font>] ) )
 <span class="kw">from</span> <span class="lab"><a class="ref" href="finseq_1.html#S5" onmouseover="rs('finseq_1/S5')" onmouseout="rh()">FINSEQ_1:sch 5</a>(<a class="txt" href="#E1:22_4_3"><span class="lab"><font color="Green" title="E47">A278</font></span></a>);<br/></span>
<span class="kw">take </span><font color="Maroon" title="c21">x</font> =  <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">Sum</a> <font color="Maroon" title="c20">q</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">S<sub>2</sub></font>[<font color="Maroon" title="c19">k</font>,<font color="Maroon" title="c21">x</font>]</span><br/>

<a NAME="E4:22_4_3"/><span class="lab"><font color="Green" title="E49">A283</font></span>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c19">k</font></span>)</span> <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> <span class="p2">(<span class="default"><font color="Maroon" title="c16">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c19">k</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="E5:22_4_3"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c20">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c19">k</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:22_4_3"><span class="lab"><font color="Green" title="E48">A282</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/>
<span class="kw">hence </span><a NAME="E6:22_4_3"/>
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c19">k</font>,<font color="Maroon" title="c21">x</font>]
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:22_4_3"><span class="lab"><font color="Green" title="E48">A282</font></span></a>, <a class="txt" href="#E4:22_4_3"><span class="lab"><font color="Green" title="E49">A283</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">consider </span><font color="Maroon" title="c19">xD</font> being   <a href="struct_0.html#NM9" title="STRUCT_0:NM.9">sequence</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"> such that </span><br/><a NAME="E13:22_4"/><span class="lab"><font color="Green" title="E47">A284</font></span>: 
 for <font color="Olive" title="b1">z</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  holds  <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">z</font>,<font color="Maroon" title="c19">xD</font> <a href="normsp_0.html#K1" title="NORMSP_0:func.1">.</a> <font color="Olive" title="b1">z</font>]
 <span class="kw">from</span> <span class="lab"><a class="ref" href="funct_2.html#S3" onmouseover="rs('funct_2/S3')" onmouseout="rh()">FUNCT_2:sch 3</a>(<a class="txt" href="#E11:22_4"><span class="lab"><font color="Green" title="E46">A277</font></span></a>);<br/></span>
<a NAME="E14:22_4"/><span class="lab"><font color="Green" title="E48">B314</font></span>: 
 for <font color="Olive" title="b1">n</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  holds  <span class="p1">(<span class="default"><font color="Maroon" title="c7">f</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c19">xD</font></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<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="c12">rho</font>,<font color="Maroon" title="c15">u</font>,<font color="Maroon" title="c16">T</font></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Olive" title="b1">n</font>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="22_4_4"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c20">n</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">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">(<span class="default"><font color="Maroon" title="c7">f</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c19">xD</font></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c20">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<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="c12">rho</font>,<font color="Maroon" title="c15">u</font>,<font color="Maroon" title="c16">T</font></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c20">n</font></span><br/>

<span class="kw">consider </span><font color="Maroon" title="c21">p1</font> being    <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> <span class="kw"> such that </span><br/><a NAME="E2:22_4_4"/><span class="lab"><font color="Green" title="E49">A285</font></span>: 
( <font color="Maroon" title="c21">p1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">F</font> <a href="finseq_2.html#K8" title="FINSEQ_2:func.8">.</a> <font color="Maroon" title="c20">n</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c21">p1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c20">n</font></span>)</span> &amp; (  for <font color="Olive" title="b1">i</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Olive" title="b1">i</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="c16">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c20">n</font></span>)</span> holds <br/>( <font color="Maroon" title="c21">p1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font> <a href="tarski.html#R2" title="TARSKI: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="c15">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="c16">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c20">n</font></span>)</span>,<font color="Olive" title="b1">i</font>)</span>)</span></span>)</span> &amp;  ex <font color="Olive" title="b2">z</font> being   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a> st <br/>( <font color="Olive" title="b2">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">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="c16">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c20">n</font></span>)</span>,<font color="Olive" title="b1">i</font>)</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c21">p1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font></span>)</span> &amp; <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="c12">rho</font>,<font color="Maroon" title="c15">u</font>,<font color="Maroon" title="c16">T</font> <a href="integr22.html#K3" title="INTEGR22:func.3">.</a> <font color="Maroon" title="c20">n</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">z</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="c16">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c20">n</font></span>)</span>,<font color="Olive" title="b1">i</font>)</span>)</span>,<font color="Maroon" title="c12">rho</font>)</span>)</span> ) ) ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:22_4"><span class="lab"><font color="Green" title="E45">A276</font></span></a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c22">q1</font> being   <a href="struct_0.html#NM7" title="STRUCT_0:NM.7">FinSequence</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"> such that </span><br/><a NAME="E4:22_4_4"/><span class="lab"><font color="Green" title="E50">A286</font></span>: 
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c22">q1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c20">n</font></span>)</span> &amp; <font color="Maroon" title="c19">xD</font> <a href="normsp_0.html#K1" title="NORMSP_0:func.1">.</a> <font color="Maroon" title="c20">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">Sum</a> <font color="Maroon" title="c22">q1</font> &amp; (  for <font color="Olive" title="b1">i</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Olive" title="b1">i</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="c16">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c20">n</font></span>)</span> holds <br/> ex <font color="Olive" title="b2">r</font> being   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a> st <br/>( <span class="p1">(<span class="default"><font color="Maroon" title="c18">F</font> <a href="finseq_2.html#K8" title="FINSEQ_2:func.8">.</a> <font color="Maroon" title="c20">n</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font> <a href="tarski.html#R2" title="TARSKI: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="c15">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="c16">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c20">n</font></span>)</span>,<font color="Olive" title="b1">i</font>)</span>)</span></span>)</span> &amp; <font color="Olive" title="b2">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">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="c16">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c20">n</font></span>)</span>,<font color="Olive" title="b1">i</font>)</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c18">F</font> <a href="finseq_2.html#K8" title="FINSEQ_2:func.8">.</a> <font color="Maroon" title="c20">n</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font></span>)</span> &amp; <font color="Maroon" title="c22">q1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">r</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="dualsp05.html#K2" title="DUALSP05:func.2">Dp1</a> (<font color="Maroon" title="c11">m</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c16">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c20">n</font></span>)</span>,<span class="p3">(<span class="default"><font color="Olive" title="b1">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>)</span>)</span> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <span class="p2">(<span class="default"><a href="dualsp05.html#K2" title="DUALSP05:func.2">Dp1</a> (<font color="Maroon" title="c11">m</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c16">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c20">n</font></span>)</span>,<font color="Olive" title="b1">i</font>)</span>)</span></span>)</span> ) ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:22_4"><span class="lab"><font color="Green" title="E47">A284</font></span></a></span>;<br/>
<a NAME="E5:22_4_4"/><span class="lab"><font color="Green" title="E51">A287</font></span>: 
<span class="p1">(<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="c12">rho</font>,<font color="Maroon" title="c15">u</font>,<font color="Maroon" title="c16">T</font></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c20">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</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="c12">rho</font>,<font color="Maroon" title="c15">u</font>,<font color="Maroon" title="c16">T</font> <a href="integr22.html#K3" title="INTEGR22:func.3">.</a> <font color="Maroon" title="c20">n</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>;<br/>
<a NAME="E6:22_4_4"/>
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c19">xD</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D1" onmouseover="rs('funct_2/D1')" onmouseout="rh()">FUNCT_2:def 1</a></span>;<br/>
<span class="kw">then </span><span class="lab"><font color="Green" title="E52">A288</font></span>: <span class="p1">(<span class="default"><font color="Maroon" title="c7">f</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c19">xD</font></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c20">n</font> = 
<font color="Maroon" title="c7">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K4" title="RLVECT_1:func.4">Sum</a> <font color="Maroon" title="c22">q1</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:22_4_4"><span class="lab"><font color="Green" title="E50">A286</font></span></a>, <a class="ref" href="funct_1.html#T13" onmouseover="rs('funct_1/T13')" onmouseout="rh()">FUNCT_1:13</a></span>
<br/>.= 
 <a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">f</font> <a href="finseqop.html#K4" title="FINSEQOP:func.4">*</a> <font color="Maroon" title="c22">q1</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="dualsp05.html#T10" target="_self" onmouseover="rs('dualsp05/T10')" onmouseout="rh()">LM87</a></span>
;<br/>
<a NAME="E8:22_4_4"/>
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c7">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="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="funct_2.html#D1" onmouseover="rs('funct_2/D1')" onmouseout="rh()">FUNCT_2:def 1</a></span>;<br/>
<a NAME="E9:22_4_4"/><span class="kw">then </span>
 <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c22">q1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c7">f</font>
 
;<br/>
<span class="kw">then </span> <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">f</font> <a href="finseqop.html#K4" title="FINSEQOP:func.4">*</a> <font color="Maroon" title="c22">q1</font></span>)</span> = 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c22">q1</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#T27" onmouseover="rs('relat_1/T27')" onmouseout="rh()">RELAT_1:27</a></span>
<br/>.= 
 <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> <font color="Maroon" title="c22">q1</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 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="c12">rho</font>,<font color="Maroon" title="c15">u</font>,<font color="Maroon" title="c16">T</font> <a href="integr22.html#K3" title="INTEGR22:func.3">.</a> <font color="Maroon" title="c20">n</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E22:22"><span class="lab"><font color="Green" title="E33">A251</font></span></a>, <a class="txt" href="#E1:22_4"><span class="lab"><font color="Green" title="E38">A260</font></span></a>, <a class="txt" href="#E4:22_4_4"><span class="lab"><font color="Green" title="E50">A286</font></span></a>, <a class="ref" href="integr22.html#D5" onmouseover="rs('integr22/D5')" onmouseout="rh()">INTEGR22:def 5</a></span>
;<br/>
<a NAME="E11:22_4_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E53">A289</font></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">f</font> <a href="finseqop.html#K4" title="FINSEQOP:func.4">*</a> <font color="Maroon" title="c22">q1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <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="c12">rho</font>,<font color="Maroon" title="c15">u</font>,<font color="Maroon" title="c16">T</font> <a href="integr22.html#K3" title="INTEGR22:func.3">.</a> <font color="Maroon" title="c20">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="E12:22_4_4"/>
 for <font color="Olive" title="b1">k</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">k</font> &amp; <font color="Olive" title="b1">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <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="c12">rho</font>,<font color="Maroon" title="c15">u</font>,<font color="Maroon" title="c16">T</font> <a href="integr22.html#K3" title="INTEGR22:func.3">.</a> <font color="Maroon" title="c20">n</font></span>)</span> holds <br/><span class="p1">(<span class="default"><font color="Maroon" title="c7">f</font> <a href="finseqop.html#K4" title="FINSEQOP:func.4">*</a> <font color="Maroon" title="c22">q1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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="c12">rho</font>,<font color="Maroon" title="c15">u</font>,<font color="Maroon" title="c16">T</font> <a href="integr22.html#K3" title="INTEGR22:func.3">.</a> <font color="Maroon" title="c20">n</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">k</font>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="22_4_4_3"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c23">k</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"> ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c23">k</font> &amp; <font color="Maroon" title="c23">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <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="c12">rho</font>,<font color="Maroon" title="c15">u</font>,<font color="Maroon" title="c16">T</font> <a href="integr22.html#K3" title="INTEGR22:func.3">.</a> <font color="Maroon" title="c20">n</font></span>)</span> implies <span class="p1">(<span class="default"><font color="Maroon" title="c7">f</font> <a href="finseqop.html#K4" title="FINSEQOP:func.4">*</a> <font color="Maroon" title="c22">q1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c23">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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="c12">rho</font>,<font color="Maroon" title="c15">u</font>,<font color="Maroon" title="c16">T</font> <a href="integr22.html#K3" title="INTEGR22:func.3">.</a> <font color="Maroon" title="c20">n</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c23">k</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:22_4_4_3"/><span class="lab"><font color="Green" title="E54">A290</font></span>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c23">k</font> &amp; <font color="Maroon" title="c23">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <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="c12">rho</font>,<font color="Maroon" title="c15">u</font>,<font color="Maroon" title="c16">T</font> <a href="integr22.html#K3" title="INTEGR22:func.3">.</a> <font color="Maroon" title="c20">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">(<span class="default"><font color="Maroon" title="c7">f</font> <a href="finseqop.html#K4" title="FINSEQOP:func.4">*</a> <font color="Maroon" title="c22">q1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c23">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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="c12">rho</font>,<font color="Maroon" title="c15">u</font>,<font color="Maroon" title="c16">T</font> <a href="integr22.html#K3" title="INTEGR22:func.3">.</a> <font color="Maroon" title="c20">n</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c23">k</font></span><br/>

<a NAME="E2:22_4_4_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E55">B291</font></span>: 
<font color="Maroon" title="c23">k</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="c12">rho</font>,<font color="Maroon" title="c15">u</font>,<font color="Maroon" title="c16">T</font> <a href="integr22.html#K3" title="INTEGR22:func.3">.</a> <font color="Maroon" title="c20">n</font></span>)</span></span>)</span>
 
;<br/>
<a NAME="E3:22_4_4_3"/><span class="kw">then </span>
<font color="Maroon" title="c23">k</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="c16">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c20">n</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E22:22"><span class="lab"><font color="Green" title="E33">A251</font></span></a>, <a class="txt" href="#E1:22_4"><span class="lab"><font color="Green" title="E38">A260</font></span></a>, <a class="ref" href="integr22.html#D5" onmouseover="rs('integr22/D5')" onmouseout="rh()">INTEGR22:def 5</a></span>;<br/>
<a NAME="E4:22_4_4_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E56">A292</font></span>: 
<font color="Maroon" title="c23">k</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="c16">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c20">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">then </span><span class="kw">consider </span><font color="Maroon" title="c24">z</font> being   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a><span class="kw"> such that </span><br/><a NAME="E6:22_4_4_3"/><span class="lab"><font color="Green" title="E57">A293</font></span>: 
( <font color="Maroon" title="c24">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">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="c16">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c20">n</font></span>)</span>,<font color="Maroon" title="c23">k</font>)</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c18">F</font> <a href="finseq_2.html#K8" title="FINSEQ_2:func.8">.</a> <font color="Maroon" title="c20">n</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c23">k</font></span>)</span> &amp; <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="c12">rho</font>,<font color="Maroon" title="c15">u</font>,<font color="Maroon" title="c16">T</font> <a href="integr22.html#K3" title="INTEGR22:func.3">.</a> <font color="Maroon" title="c20">n</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c23">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">z</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="c16">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c20">n</font></span>)</span>,<font color="Maroon" title="c23">k</font>)</span>)</span>,<font color="Maroon" title="c12">rho</font>)</span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:22_4_4"><span class="lab"><font color="Green" title="E49">A285</font></span></a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c25">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="E8:22_4_4_3"/><span class="lab"><font color="Green" title="E58">A294</font></span>: 
( <span class="p1">(<span class="default"><font color="Maroon" title="c18">F</font> <a href="finseq_2.html#K8" title="FINSEQ_2:func.8">.</a> <font color="Maroon" title="c20">n</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c23">k</font> <a href="tarski.html#R2" title="TARSKI: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="c15">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="c16">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c20">n</font></span>)</span>,<font color="Maroon" title="c23">k</font>)</span>)</span></span>)</span> &amp; <font color="Maroon" title="c25">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">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="c16">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c20">n</font></span>)</span>,<font color="Maroon" title="c23">k</font>)</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c18">F</font> <a href="finseq_2.html#K8" title="FINSEQ_2:func.8">.</a> <font color="Maroon" title="c20">n</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c23">k</font></span>)</span> &amp; <font color="Maroon" title="c22">q1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c23">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c25">r</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="dualsp05.html#K2" title="DUALSP05:func.2">Dp1</a> (<font color="Maroon" title="c11">m</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c16">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c20">n</font></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon" title="c23">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>)</span>)</span> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <span class="p2">(<span class="default"><a href="dualsp05.html#K2" title="DUALSP05:func.2">Dp1</a> (<font color="Maroon" title="c11">m</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c16">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c20">n</font></span>)</span>,<font color="Maroon" title="c23">k</font>)</span>)</span></span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:22_4_4"><span class="lab"><font color="Green" title="E50">A286</font></span></a>, <a class="txt" href="#E4:22_4_4_3"><span class="lab"><font color="Green" title="E56">A292</font></span></a></span>;<br/>
<a NAME="E9:22_4_4_3"/>
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c23">k</font> &amp; <font color="Maroon" title="c23">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c20">n</font></span>)</span> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E22:22"><span class="lab"><font color="Green" title="E33">A251</font></span></a>, <a class="txt" href="#E1:22_4"><span class="lab"><font color="Green" title="E38">A260</font></span></a>, <a class="txt" href="#E1:22_4_4_3"><span class="lab"><font color="Green" title="E54">A290</font></span></a>, <a class="ref" href="integr22.html#D5" onmouseover="rs('integr22/D5')" onmouseout="rh()">INTEGR22:def 5</a></span>;<br/>
<a NAME="E10:22_4_4_3"/><span class="kw">then </span>
<font color="Maroon" title="c23">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</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="c16">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c20">n</font></span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T7" onmouseover="rs('xreal_1/T7')" onmouseout="rh()">XREAL_1:7</a></span>;<br/>
<a NAME="E11:22_4_4_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E59">A296</font></span>: 
<font color="Maroon" title="c23">k</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"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c16">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c20">n</font></span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:22_4_4_3"><span class="lab"><font color="Green" title="E54">A290</font></span></a></span>;<br/>
<a NAME="E12:22_4_4_3"/><span class="lab"><font color="Green" title="E60">d</font></span>: 
( 1 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c23">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <font color="Maroon" title="c23">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c20">n</font></span>)</span> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E22:22"><span class="lab"><font color="Green" title="E33">A251</font></span></a>, <a class="txt" href="#E1:22_4"><span class="lab"><font color="Green" title="E38">A260</font></span></a>, <a class="txt" href="#E1:22_4_4_3"><span class="lab"><font color="Green" title="E54">A290</font></span></a>, <a class="ref" href="integr22.html#D5" onmouseover="rs('integr22/D5')" onmouseout="rh()">INTEGR22:def 5</a>, <a class="ref" href="xreal_1.html#T7" onmouseover="rs('xreal_1/T7')" onmouseout="rh()">XREAL_1:7</a></span>;<br/>
<a NAME="E13:22_4_4_3"/><span class="kw">then </span>
<font color="Maroon" title="c23">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</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="c16">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c20">n</font></span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T7" onmouseover="rs('xreal_1/T7')" onmouseout="rh()">XREAL_1:7</a></span>;<br/>
<a NAME="E14:22_4_4_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E61">A297</font></span>: 
<font color="Maroon" title="c23">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <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"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c16">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c20">n</font></span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:22_4_4_3"><span class="lab"><font color="Green" title="E60">d</font></span></a></span>;<br/>
<a NAME="E15:22_4_4_3"/>
<font color="Maroon" title="c23">k</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> <font color="Maroon" title="c22">q1</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:22_4_4"><span class="lab"><font color="Green" title="E50">A286</font></span></a>, <a class="txt" href="#E2:22_4_4_3"><span class="lab"><font color="Green" title="E55">B291</font></span></a>, <a class="txt" href="#E22:22"><span class="lab"><font color="Green" title="E33">A251</font></span></a>, <a class="txt" href="#E1:22_4"><span class="lab"><font color="Green" title="E38">A260</font></span></a>, <a class="ref" href="integr22.html#D5" onmouseover="rs('integr22/D5')" onmouseout="rh()">INTEGR22:def 5</a></span>;<br/>
<a NAME="E16:22_4_4_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E62">A298</font></span>: 
<font color="Maroon" title="c23">k</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> <font color="Maroon" title="c22">q1</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/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="22_4_4_3_1"><span class="kw">per </span></a><a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:22_4_4_3_1"/>
( <font color="Maroon" title="c23">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or <font color="Maroon" title="c23">k</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 1 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="22_4_4_3_1_1"><span class="kw">suppose </span></a><a NAME="E1:22_4_4_3_1_1"/><span class="lab"><font color="Green" title="E63">A299</font></span>: 
<font color="Maroon" title="c23">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c7">f</font> <a href="finseqop.html#K4" title="FINSEQOP:func.4">*</a> <font color="Maroon" title="c22">q1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c23">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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="c12">rho</font>,<font color="Maroon" title="c15">u</font>,<font color="Maroon" title="c16">T</font> <a href="integr22.html#K3" title="INTEGR22:func.3">.</a> <font color="Maroon" title="c20">n</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c23">k</font></span><br/><div class="add"><a NAME="E2:22_4_4_3_1_1"/>
<font color="Maroon" title="c1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><span class="p2">(<span class="default"><a href="seq_4.html#K3" title="SEQ_4:func.3">lower_bound</a> <font color="Maroon" title="c1">A</font></span>)</span>,<span class="p2">(<span class="default"><a href="seq_4.html#K2" title="SEQ_4:func.2">upper_bound</a> <font color="Maroon" title="c1">A</font></span>)</span></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="integra1.html#T4" onmouseover="rs('integra1/T4')" onmouseout="rh()">INTEGRA1:4</a></span>;<br/><a NAME="E3:22_4_4_3_1_1"/><span class="kw">then </span>
 <a href="seq_4.html#K3" title="SEQ_4:func.3">lower_bound</a> <font color="Maroon" title="c1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">a</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:22"><span class="lab"><font color="Green" title="E28">A7</font></span></a>, <a class="ref" href="integra1.html#T5" onmouseover="rs('integra1/T5')" onmouseout="rh()">INTEGRA1:5</a></span>;<br/><a NAME="E4:22_4_4_3_1_1"/><span class="kw">then </span>
 <a href="seq_4.html#K3" title="SEQ_4:func.3">lower_bound</a> <font color="Maroon" title="c1">A</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="#E11:22"><span class="lab"><font color="Green" title="E28">A7</font></span></a></span>;<br/><a NAME="E5:22_4_4_3_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E64">A301</font></span>: 
 <a href="seq_4.html#K3" title="SEQ_4:func.3">lower_bound</a> <font color="Maroon" title="c1">A</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> <font color="Maroon" title="c11">m</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D1" onmouseover="rs('funct_2/D1')" onmouseout="rh()">FUNCT_2:def 1</a></span>;<br/><a NAME="E6:22_4_4_3_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c16">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c20">n</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c23">k</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c1">A</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:22_4_4_3"><span class="lab"><font color="Green" title="E56">A292</font></span></a>, <a class="ref" href="integra1.html#T6" onmouseover="rs('integra1/T6')" onmouseout="rh()">INTEGRA1:6</a></span>;<br/><a NAME="E7:22_4_4_3_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E65">A302</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c16">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c20">n</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c23">k</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c11">m</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D1" onmouseover="rs('funct_2/D1')" onmouseout="rh()">FUNCT_2:def 1</a></span>;<br/><a NAME="E8:22_4_4_3_1_1"/><span class="lab"><font color="Green" title="E66">A303</font></span>: 
(  <a href="seq_4.html#K3" title="SEQ_4:func.3">lower_bound</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c16">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c20">n</font></span>)</span>,<font color="Maroon" title="c23">k</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="seq_4.html#K3" title="SEQ_4:func.3">lower_bound</a> <font color="Maroon" title="c1">A</font> &amp;  <a href="seq_4.html#K2" title="SEQ_4:func.2">upper_bound</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c16">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c20">n</font></span>)</span>,<font color="Maroon" title="c23">k</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c20">n</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c23">k</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:22_4_4_3"><span class="lab"><font color="Green" title="E56">A292</font></span></a>, <a class="txt" href="#E1:22_4_4_3_1_1"><span class="lab"><font color="Green" title="E63">A299</font></span></a>, <a class="ref" href="integra1.html#D4" onmouseover="rs('integra1/D4')" onmouseout="rh()">INTEGRA1:def 4</a></span>;<br/><span class="lab"><font color="Green" title="E67">A304</font></span>:  <a href="dualsp05.html#K2" title="DUALSP05:func.2">Dp1</a> (<font color="Maroon" title="c11">m</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c16">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c20">n</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c23">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>) = 
<font color="Maroon" title="c11">m</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c16">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c20">n</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c23">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:22_4_4_3"><span class="lab"><font color="Green" title="E61">A297</font></span></a>, <a class="ref" href="dualsp05.html#D2" target="_self" onmouseover="rs('dualsp05/D2')" onmouseout="rh()">defDp1</a>, <a class="txt" href="#E1:22_4_4_3_1_1"><span class="lab"><font color="Green" title="E63">A299</font></span></a></span>
<br/>.= 
<font color="Maroon" title="c11">m</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c16">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c20">n</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c23">k</font></span>)</span>

;<br/><span class="lab"><font color="Green" title="E68">A306</font></span>: <font color="Maroon" title="c7">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="dualsp05.html#K2" title="DUALSP05:func.2">Dp1</a> (<font color="Maroon" title="c11">m</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c16">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c20">n</font></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon" title="c23">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>)</span>)</span> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <span class="p2">(<span class="default"><a href="dualsp05.html#K2" title="DUALSP05:func.2">Dp1</a> (<font color="Maroon" title="c11">m</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c16">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c20">n</font></span>)</span>,<font color="Maroon" title="c23">k</font>)</span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c7">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><a href="dualsp05.html#K2" title="DUALSP05:func.2">Dp1</a> (<font color="Maroon" title="c11">m</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c16">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c20">n</font></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon" title="c23">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>)</span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><a href="dualsp05.html#K2" title="DUALSP05:func.2">Dp1</a> (<font color="Maroon" title="c11">m</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c16">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c20">n</font></span>)</span>,<font color="Maroon" title="c23">k</font>)</span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="hahnban.html#T19" onmouseover="rs('hahnban/T19')" onmouseout="rh()">HAHNBAN:19</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c7">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c11">m</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c16">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c20">n</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c23">k</font></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c11">m</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><a href="seq_4.html#K3" title="SEQ_4:func.3">lower_bound</a> <font color="Maroon" title="c1">A</font></span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:22_4_4_3_1_1"><span class="lab"><font color="Green" title="E67">A304</font></span></a>, <a class="txt" href="#E11:22_4_4_3"><span class="lab"><font color="Green" title="E59">A296</font></span></a>, <a class="txt" href="#E1:22_4_4_3_1_1"><span class="lab"><font color="Green" title="E63">A299</font></span></a>, <a class="ref" href="dualsp05.html#D2" target="_self" onmouseover="rs('dualsp05/D2')" onmouseout="rh()">defDp1</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">f</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c11">m</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c16">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c20">n</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c23">k</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c11">m</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><a href="seq_4.html#K3" title="SEQ_4:func.3">lower_bound</a> <font color="Maroon" title="c1">A</font></span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:22_4_4_3_1_1"><span class="lab"><font color="Green" title="E65">A302</font></span></a>, <a class="ref" href="funct_1.html#T13" onmouseover="rs('funct_1/T13')" onmouseout="rh()">FUNCT_1:13</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c12">rho</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><a href="seq_4.html#K2" title="SEQ_4:func.2">upper_bound</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="c16">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c20">n</font></span>)</span>,<font color="Maroon" title="c23">k</font>)</span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">rho</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><a href="seq_4.html#K3" title="SEQ_4:func.3">lower_bound</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="c16">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c20">n</font></span>)</span>,<font color="Maroon" title="c23">k</font>)</span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:22_4_4_3_1_1"><span class="lab"><font color="Green" title="E66">A303</font></span></a>, <a class="txt" href="#E5:22_4_4_3_1_1"><span class="lab"><font color="Green" title="E64">A301</font></span></a>, <a class="ref" href="funct_1.html#T13" onmouseover="rs('funct_1/T13')" onmouseout="rh()">FUNCT_1:13</a></span>
<br/>.= 
 <a href="integr22.html#K1" title="INTEGR22:func.1">vol</a> (<span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c16">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c20">n</font></span>)</span>,<font color="Maroon" title="c23">k</font>)</span>)</span>,<font color="Maroon" title="c12">rho</font>)
<span class="kw">by</span> <span class="lab"><a class="ref" href="integr22.html#D1" onmouseover="rs('integr22/D1')" onmouseout="rh()">INTEGR22:def 1</a></span>
;<br/><span class="kw">thus </span><span class="p1">(<span class="default"><font color="Maroon" title="c7">f</font> <a href="finseqop.html#K4" title="FINSEQOP:func.4">*</a> <font color="Maroon" title="c22">q1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c23">k</font> = 
<font color="Maroon" title="c7">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c25">r</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="dualsp05.html#K2" title="DUALSP05:func.2">Dp1</a> (<font color="Maroon" title="c11">m</font>,<span class="p4">(<span class="default"><font color="Maroon" title="c16">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c20">n</font></span>)</span>,<span class="p4">(<span class="default"><font color="Maroon" title="c23">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>)</span>)</span> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <span class="p3">(<span class="default"><a href="dualsp05.html#K2" title="DUALSP05:func.2">Dp1</a> (<font color="Maroon" title="c11">m</font>,<span class="p4">(<span class="default"><font color="Maroon" title="c16">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c20">n</font></span>)</span>,<font color="Maroon" title="c23">k</font>)</span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:22_4_4_3"><span class="lab"><font color="Green" title="E58">A294</font></span></a>, <a class="txt" href="#E16:22_4_4_3"><span class="lab"><font color="Green" title="E62">A298</font></span></a>, <a class="ref" href="funct_1.html#T13" onmouseover="rs('funct_1/T13')" onmouseout="rh()">FUNCT_1:13</a></span>
<br/>.= 
<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="c12">rho</font>,<font color="Maroon" title="c15">u</font>,<font color="Maroon" title="c16">T</font> <a href="integr22.html#K3" title="INTEGR22:func.3">.</a> <font color="Maroon" title="c20">n</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c23">k</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:22_4_4_3"><span class="lab"><font color="Green" title="E57">A293</font></span></a>, <a class="txt" href="#E8:22_4_4_3"><span class="lab"><font color="Green" title="E58">A294</font></span></a>, <a class="txt" href="#E10:22_4_4_3_1_1"><span class="lab"><font color="Green" title="E68">A306</font></span></a>, <a class="ref" href="hahnban.html#D3" onmouseover="rs('hahnban/D3')" onmouseout="rh()">HAHNBAN:def 3</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><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="22_4_4_3_1_2"><span class="kw">suppose </span></a><a NAME="E1:22_4_4_3_1_2"/><span class="lab"><font color="Green" title="E63">A307</font></span>: 
<font color="Maroon" title="c23">k</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 1
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c7">f</font> <a href="finseqop.html#K4" title="FINSEQOP:func.4">*</a> <font color="Maroon" title="c22">q1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c23">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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="c12">rho</font>,<font color="Maroon" title="c15">u</font>,<font color="Maroon" title="c16">T</font> <a href="integr22.html#K3" title="INTEGR22:func.3">.</a> <font color="Maroon" title="c20">n</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c23">k</font></span><br/><div class="add"><a NAME="E2:22_4_4_3_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c16">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c20">n</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c23">k</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c1">A</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:22_4_4_3"><span class="lab"><font color="Green" title="E56">A292</font></span></a>, <a class="ref" href="integra1.html#T6" onmouseover="rs('integra1/T6')" onmouseout="rh()">INTEGRA1:6</a></span>;<br/><a NAME="E3:22_4_4_3_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E64">A308</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c16">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c20">n</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c23">k</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c11">m</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D1" onmouseover="rs('funct_2/D1')" onmouseout="rh()">FUNCT_2:def 1</a></span>;<br/><a NAME="E4:22_4_4_3_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c16">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c20">n</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c23">k</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c1">A</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:22_4_4_3"><span class="lab"><font color="Green" title="E56">A292</font></span></a>, <a class="txt" href="#E1:22_4_4_3_1_2"><span class="lab"><font color="Green" title="E63">A307</font></span></a>, <a class="ref" href="integra1.html#T7" onmouseover="rs('integra1/T7')" onmouseout="rh()">INTEGRA1:7</a></span>;<br/><a NAME="E5:22_4_4_3_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E65">A309</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c16">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c20">n</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c23">k</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c11">m</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D1" onmouseover="rs('funct_2/D1')" onmouseout="rh()">FUNCT_2:def 1</a></span>;<br/><a NAME="E6:22_4_4_3_1_2"/><span class="lab"><font color="Green" title="E66">A310</font></span>: 
(  <a href="seq_4.html#K3" title="SEQ_4:func.3">lower_bound</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c16">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c20">n</font></span>)</span>,<font color="Maroon" title="c23">k</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c20">n</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c23">k</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span> &amp;  <a href="seq_4.html#K2" title="SEQ_4:func.2">upper_bound</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c16">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c20">n</font></span>)</span>,<font color="Maroon" title="c23">k</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c20">n</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c23">k</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:22_4_4_3"><span class="lab"><font color="Green" title="E56">A292</font></span></a>, <a class="txt" href="#E1:22_4_4_3_1_2"><span class="lab"><font color="Green" title="E63">A307</font></span></a>, <a class="ref" href="integra1.html#D4" onmouseover="rs('integra1/D4')" onmouseout="rh()">INTEGRA1:def 4</a></span>;<br/><a NAME="E7:22_4_4_3_1_2"/>
1 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c23">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T8" onmouseover="rs('xreal_1/T8')" onmouseout="rh()">XREAL_1:8</a>, <a class="txt" href="#E1:22_4_4_3"><span class="lab"><font color="Green" title="E54">A290</font></span></a></span>;<br/><span class="kw">then </span><span class="lab"><font color="Green" title="E67">A311</font></span>:  <a href="dualsp05.html#K2" title="DUALSP05:func.2">Dp1</a> (<font color="Maroon" title="c11">m</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c16">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c20">n</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c23">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>) = 
<font color="Maroon" title="c11">m</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c16">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c20">n</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c23">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:22_4_4_3"><span class="lab"><font color="Green" title="E61">A297</font></span></a>, <a class="ref" href="dualsp05.html#D2" target="_self" onmouseover="rs('dualsp05/D2')" onmouseout="rh()">defDp1</a></span>
<br/>.= 
<font color="Maroon" title="c11">m</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c16">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c20">n</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c23">k</font></span>)</span>

;<br/><span class="lab"><font color="Green" title="E68">A313</font></span>: <font color="Maroon" title="c7">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="dualsp05.html#K2" title="DUALSP05:func.2">Dp1</a> (<font color="Maroon" title="c11">m</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c16">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c20">n</font></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon" title="c23">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>)</span>)</span> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <span class="p2">(<span class="default"><a href="dualsp05.html#K2" title="DUALSP05:func.2">Dp1</a> (<font color="Maroon" title="c11">m</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c16">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c20">n</font></span>)</span>,<font color="Maroon" title="c23">k</font>)</span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c7">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><a href="dualsp05.html#K2" title="DUALSP05:func.2">Dp1</a> (<font color="Maroon" title="c11">m</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c16">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c20">n</font></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon" title="c23">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>)</span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><a href="dualsp05.html#K2" title="DUALSP05:func.2">Dp1</a> (<font color="Maroon" title="c11">m</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c16">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c20">n</font></span>)</span>,<font color="Maroon" title="c23">k</font>)</span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="hahnban.html#T19" onmouseover="rs('hahnban/T19')" onmouseout="rh()">HAHNBAN:19</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c7">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c11">m</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c16">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c20">n</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c23">k</font></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c11">m</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c16">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c20">n</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c23">k</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span></span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:22_4_4_3_1_2"><span class="lab"><font color="Green" title="E67">A311</font></span></a>, <a class="txt" href="#E11:22_4_4_3"><span class="lab"><font color="Green" title="E59">A296</font></span></a>, <a class="txt" href="#E1:22_4_4_3_1_2"><span class="lab"><font color="Green" title="E63">A307</font></span></a>, <a class="ref" href="dualsp05.html#D2" target="_self" onmouseover="rs('dualsp05/D2')" onmouseout="rh()">defDp1</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">f</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c11">m</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c16">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c20">n</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c23">k</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c11">m</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c16">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c20">n</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c23">k</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span></span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:22_4_4_3_1_2"><span class="lab"><font color="Green" title="E64">A308</font></span></a>, <a class="ref" href="funct_1.html#T13" onmouseover="rs('funct_1/T13')" onmouseout="rh()">FUNCT_1:13</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c12">rho</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><a href="seq_4.html#K2" title="SEQ_4:func.2">upper_bound</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="c16">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c20">n</font></span>)</span>,<font color="Maroon" title="c23">k</font>)</span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">rho</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><a href="seq_4.html#K3" title="SEQ_4:func.3">lower_bound</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="c16">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c20">n</font></span>)</span>,<font color="Maroon" title="c23">k</font>)</span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:22_4_4_3_1_2"><span class="lab"><font color="Green" title="E66">A310</font></span></a>, <a class="txt" href="#E5:22_4_4_3_1_2"><span class="lab"><font color="Green" title="E65">A309</font></span></a>, <a class="ref" href="funct_1.html#T13" onmouseover="rs('funct_1/T13')" onmouseout="rh()">FUNCT_1:13</a></span>
<br/>.= 
 <a href="integr22.html#K1" title="INTEGR22:func.1">vol</a> (<span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c16">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c20">n</font></span>)</span>,<font color="Maroon" title="c23">k</font>)</span>)</span>,<font color="Maroon" title="c12">rho</font>)
<span class="kw">by</span> <span class="lab"><a class="ref" href="integr22.html#D1" onmouseover="rs('integr22/D1')" onmouseout="rh()">INTEGR22:def 1</a></span>
;<br/><span class="kw">thus </span><span class="p1">(<span class="default"><font color="Maroon" title="c7">f</font> <a href="finseqop.html#K4" title="FINSEQOP:func.4">*</a> <font color="Maroon" title="c22">q1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c23">k</font> = 
<font color="Maroon" title="c7">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c25">r</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="dualsp05.html#K2" title="DUALSP05:func.2">Dp1</a> (<font color="Maroon" title="c11">m</font>,<span class="p4">(<span class="default"><font color="Maroon" title="c16">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c20">n</font></span>)</span>,<span class="p4">(<span class="default"><font color="Maroon" title="c23">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>)</span>)</span> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <span class="p3">(<span class="default"><a href="dualsp05.html#K2" title="DUALSP05:func.2">Dp1</a> (<font color="Maroon" title="c11">m</font>,<span class="p4">(<span class="default"><font color="Maroon" title="c16">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c20">n</font></span>)</span>,<font color="Maroon" title="c23">k</font>)</span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:22_4_4_3"><span class="lab"><font color="Green" title="E58">A294</font></span></a>, <a class="txt" href="#E16:22_4_4_3"><span class="lab"><font color="Green" title="E62">A298</font></span></a>, <a class="ref" href="funct_1.html#T13" onmouseover="rs('funct_1/T13')" onmouseout="rh()">FUNCT_1:13</a></span>
<br/>.= 
<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="c12">rho</font>,<font color="Maroon" title="c15">u</font>,<font color="Maroon" title="c16">T</font> <a href="integr22.html#K3" title="INTEGR22:func.3">.</a> <font color="Maroon" title="c20">n</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c23">k</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:22_4_4_3"><span class="lab"><font color="Green" title="E57">A293</font></span></a>, <a class="txt" href="#E8:22_4_4_3"><span class="lab"><font color="Green" title="E58">A294</font></span></a>, <a class="txt" href="#E9:22_4_4_3_1_2"><span class="lab"><font color="Green" title="E68">A313</font></span></a>, <a class="ref" href="hahnban.html#D3" onmouseover="rs('hahnban/D3')" onmouseout="rh()">HAHNBAN:def 3</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></div><span class="kw">end;</span></div>

</div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E13:22_4_4"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c7">f</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c19">xD</font></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c20">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<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="c12">rho</font>,<font color="Maroon" title="c15">u</font>,<font color="Maroon" title="c16">T</font></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c20">n</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:22_4_4"><span class="lab"><font color="Green" title="E51">A287</font></span></a>, <a class="txt" href="#E7:22_4_4"><span class="lab"><font color="Green" title="E52">A288</font></span></a>, <a class="txt" href="#E11:22_4_4"><span class="lab"><font color="Green" title="E53">A289</font></span></a>, <a class="ref" href="finseq_1.html#T14" onmouseover="rs('finseq_1/T14')" onmouseout="rh()">FINSEQ_1:14</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="E15:22_4"/><span class="lab"><font color="Green" title="E49">A315</font></span>: 
(  <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="c12">rho</font>,<font color="Maroon" title="c15">u</font>,<font color="Maroon" title="c16">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="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="c12">rho</font>,<font color="Maroon" title="c15">u</font>,<font color="Maroon" title="c16">T</font></span>)</span> <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="c15">u</font>,<font color="Maroon" title="c12">rho</font>) )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:22_4"><span class="lab"><font color="Green" title="E40">B262</font></span></a>, <a class="txt" href="#E22:22"><span class="lab"><font color="Green" title="E33">A251</font></span></a>, <a class="txt" href="#E1:22_4"><span class="lab"><font color="Green" title="E38">A260</font></span></a>, <a class="ref" href="integr22.html#D9" onmouseover="rs('integr22/D9')" onmouseout="rh()">INTEGR22:def 9</a>, <a class="txt" href="#E5:22_4"><span class="lab"><font color="Green" title="E41">A263</font></span></a></span>;<br/>
<a NAME="E16:22_4"/><span class="lab"><font color="Green" title="E50">A316</font></span>: 
<font color="Maroon" title="c15">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="#E7:22_4"><span class="lab"><font color="Green" title="E43">A269</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">reconsider </span><font color="Maroon" title="c20">v</font> = <font color="Maroon" title="c15">u</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="txt" href="#E7:22_4"><span class="lab"><font color="Green" title="E43">A269</font></span></a>, <a class="ref" href="dualsp05.html#T7" target="_self" onmouseover="rs('dualsp05/T7')" onmouseout="rh()">Lm2</a></span>;<br/>
<a NAME="E18:22_4"/>
<font color="Maroon" title="c20">v</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="c0sp1.html#K6" title="C0SP1:func.6">BoundedFunctions</a> <font color="Maroon" title="c1">A</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:22_4"><span class="lab"><font color="Green" title="E50">A316</font></span></a>, <a class="ref" href="dualsp05.html#T5" target="_self" onmouseover="rs('dualsp05/T5')" onmouseout="rh()">Lm1</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c21">g</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><span class="kw"> such that </span><br/><a NAME="E20:22_4"/><span class="lab"><font color="Green" title="E51">A317</font></span>: 
( <font color="Maroon" title="c21">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">v</font> &amp; <font color="Maroon" title="c21">g</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c1">A</font> is  <a href="comseq_2.html#V1" title="COMSEQ_2:attr.1">bounded</a>  )
 ;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c22">v0</font> = <font color="Maroon" title="c20">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> ;<br/>
<a NAME="E22:22_4"/><span class="lab"><font color="Green" title="E52">A318</font></span>: 
 for <font color="Olive" title="b1">r</font> being   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a>  st  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">r</font> holds <br/> ex <font color="Olive" title="b2">m0</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> st <br/> for <font color="Olive" title="b3">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Olive" title="b2">m0</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b3">n</font> holds <br/><span class="p1"><a href="normsp_0.html#K2" title="NORMSP_0:func.2">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c19">xD</font> <a href="normsp_0.html#K1" title="NORMSP_0:func.1">.</a> <font color="Olive" title="b3">n</font></span>)</span> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <font color="Maroon" title="c22">v0</font></span>)</span></span><a href="normsp_0.html#K2" title="NORMSP_0:func.2">.||</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">r</font>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="22_4_5"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c23">r</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"> (  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c23">r</font> implies  ex <font color="Olive" title="b1">m0</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> st <br/> for <font color="Olive" title="b2">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Olive" title="b1">m0</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b2">n</font> holds <br/><span class="p1"><a href="normsp_0.html#K2" title="NORMSP_0:func.2">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c19">xD</font> <a href="normsp_0.html#K1" title="NORMSP_0:func.1">.</a> <font color="Olive" title="b2">n</font></span>)</span> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <font color="Maroon" title="c22">v0</font></span>)</span></span><a href="normsp_0.html#K2" title="NORMSP_0:func.2">.||</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c23">r</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:22_4_5"/><span class="lab"><font color="Green" title="E53">A319</font></span>: 
 <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c23">r</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">  ex <font color="Olive" title="b1">m0</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> st <br/> for <font color="Olive" title="b2">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Olive" title="b1">m0</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b2">n</font> holds <br/><span class="p1"><a href="normsp_0.html#K2" title="NORMSP_0:func.2">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c19">xD</font> <a href="normsp_0.html#K1" title="NORMSP_0:func.1">.</a> <font color="Olive" title="b2">n</font></span>)</span> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <font color="Maroon" title="c22">v0</font></span>)</span></span><a href="normsp_0.html#K2" title="NORMSP_0:func.2">.||</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c23">r</font></span><br/>

<a NAME="E2:22_4_5"/><span class="kw">then </span><span class="lab"><font color="Green" title="E54">A321</font></span>: 
<font color="Maroon" title="c23">r</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c23">r</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T216" onmouseover="rs('xreal_1/T216')" onmouseout="rh()">XREAL_1:216</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c24">s</font> being   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a><span class="kw"> such that </span><br/><a NAME="E4:22_4_5"/><span class="lab"><font color="Green" title="E55">A322</font></span>: 
 <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c24">s</font>
 <span class="kw">and </span><br/><a NAME="E5:22_4_5"/><span class="lab"><font color="Green" title="E56">A323</font></span>: 
 for <font color="Olive" title="b1">x1</font>, <font color="Olive" title="b2">x2</font> being   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a>  st <font color="Olive" title="b1">x1</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="c15">u</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c1">A</font></span>)</span> &amp; <font color="Olive" title="b2">x2</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="c15">u</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c1">A</font></span>)</span> &amp; <span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b1">x1</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Olive" title="b2">x2</font></span>)</span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c24">s</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"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c15">u</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x1</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c15">u</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">x2</font></span>)</span></span>)</span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c23">r</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:22_4"><span class="lab"><font color="Green" title="E39">A261</font></span></a>, <a class="txt" href="#E1:22_4_5"><span class="lab"><font color="Green" title="E53">A319</font></span></a>, <a class="ref" href="xreal_1.html#T215" onmouseover="rs('xreal_1/T215')" onmouseout="rh()">XREAL_1:215</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c25">m0</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a><span class="kw"> such that </span><br/><a NAME="E7:22_4_5"/><span class="lab"><font color="Green" title="E57">A324</font></span>: 
 for <font color="Olive" title="b1">i</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Maroon" title="c25">m0</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">i</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"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="integra3.html#K2" title="INTEGRA3:func.2">delta</a> <font color="Maroon" title="c16">T</font></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Olive" title="b1">i</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span>)</span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c24">s</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:22_4"><span class="lab"><font color="Green" title="E41">A263</font></span></a>, <a class="txt" href="#E4:22_4_5"><span class="lab"><font color="Green" title="E55">A322</font></span></a>, <a class="ref" href="seq_2.html#D7" onmouseover="rs('seq_2/D7')" onmouseout="rh()">SEQ_2:def 7</a></span>;<br/>
<a NAME="E8:22_4_5"/><span class="lab"><font color="Green" title="E58">A325</font></span>: 
 for <font color="Olive" title="b1">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Maroon" title="c25">m0</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">n</font> holds <br/><span class="p1"><a href="normsp_0.html#K2" title="NORMSP_0:func.2">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c19">xD</font> <a href="normsp_0.html#K1" title="NORMSP_0:func.1">.</a> <font color="Olive" title="b1">n</font></span>)</span> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <font color="Maroon" title="c22">v0</font></span>)</span></span><a href="normsp_0.html#K2" title="NORMSP_0:func.2">.||</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c23">r</font>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="22_4_5_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c26">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"> ( <font color="Maroon" title="c25">m0</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c26">n</font> implies <span class="p1"><a href="normsp_0.html#K2" title="NORMSP_0:func.2">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c19">xD</font> <a href="normsp_0.html#K1" title="NORMSP_0:func.1">.</a> <font color="Maroon" title="c26">n</font></span>)</span> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <font color="Maroon" title="c22">v0</font></span>)</span></span><a href="normsp_0.html#K2" title="NORMSP_0:func.2">.||</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c23">r</font> )</span><br/>

<a NAME="E1:22_4_5_1"/><span class="lab"><font color="Green" title="E59">A326</font></span>: 
<font color="Maroon" title="c26">n</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal1.html#D12" onmouseover="rs('ordinal1/D12')" onmouseout="rh()">ORDINAL1:def 12</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c27">p2</font> being    <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> <span class="kw"> such that </span><br/><a NAME="E3:22_4_5_1"/><span class="lab"><font color="Green" title="E60">A327</font></span>: 
( <font color="Maroon" title="c27">p2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">F</font> <a href="finseq_2.html#K8" title="FINSEQ_2:func.8">.</a> <font color="Maroon" title="c26">n</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c27">p2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c26">n</font></span>)</span> &amp; (  for <font color="Olive" title="b1">i</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Olive" title="b1">i</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="c16">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c26">n</font></span>)</span> holds <br/>( <font color="Maroon" title="c27">p2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font> <a href="tarski.html#R2" title="TARSKI: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="c15">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="c16">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c26">n</font></span>)</span>,<font color="Olive" title="b1">i</font>)</span>)</span></span>)</span> &amp;  ex <font color="Olive" title="b2">z</font> being   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a> st <br/>( <font color="Olive" title="b2">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">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="c16">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c26">n</font></span>)</span>,<font color="Olive" title="b1">i</font>)</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c27">p2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font></span>)</span> &amp; <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="c12">rho</font>,<font color="Maroon" title="c15">u</font>,<font color="Maroon" title="c16">T</font> <a href="integr22.html#K3" title="INTEGR22:func.3">.</a> <font color="Maroon" title="c26">n</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">z</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="c16">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c26">n</font></span>)</span>,<font color="Olive" title="b1">i</font>)</span>)</span>,<font color="Maroon" title="c12">rho</font>)</span>)</span> ) ) ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:22_4"><span class="lab"><font color="Green" title="E45">A276</font></span></a>, <a class="txt" href="#E1:22_4_5_1"><span class="lab"><font color="Green" title="E59">A326</font></span></a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c28">q2</font> being   <a href="struct_0.html#NM7" title="STRUCT_0:NM.7">FinSequence</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"> such that </span><br/><a NAME="E5:22_4_5_1"/><span class="lab"><font color="Green" title="E61">A328</font></span>: 
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c28">q2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c26">n</font></span>)</span> &amp; <font color="Maroon" title="c19">xD</font> <a href="normsp_0.html#K1" title="NORMSP_0:func.1">.</a> <font color="Maroon" title="c26">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">Sum</a> <font color="Maroon" title="c28">q2</font> &amp; (  for <font color="Olive" title="b1">i</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Olive" title="b1">i</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="c16">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c26">n</font></span>)</span> holds <br/> ex <font color="Olive" title="b2">r</font> being   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a> st <br/>( <span class="p1">(<span class="default"><font color="Maroon" title="c18">F</font> <a href="finseq_2.html#K8" title="FINSEQ_2:func.8">.</a> <font color="Maroon" title="c26">n</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font> <a href="tarski.html#R2" title="TARSKI: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="c15">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="c16">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c26">n</font></span>)</span>,<font color="Olive" title="b1">i</font>)</span>)</span></span>)</span> &amp; <font color="Olive" title="b2">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">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="c16">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c26">n</font></span>)</span>,<font color="Olive" title="b1">i</font>)</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c18">F</font> <a href="finseq_2.html#K8" title="FINSEQ_2:func.8">.</a> <font color="Maroon" title="c26">n</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font></span>)</span> &amp; <font color="Maroon" title="c28">q2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">r</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="dualsp05.html#K2" title="DUALSP05:func.2">Dp1</a> (<font color="Maroon" title="c11">m</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c16">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c26">n</font></span>)</span>,<span class="p3">(<span class="default"><font color="Olive" title="b1">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>)</span>)</span> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <span class="p2">(<span class="default"><a href="dualsp05.html#K2" title="DUALSP05:func.2">Dp1</a> (<font color="Maroon" title="c11">m</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c16">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c26">n</font></span>)</span>,<font color="Olive" title="b1">i</font>)</span>)</span></span>)</span> ) ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:22_4"><span class="lab"><font color="Green" title="E47">A284</font></span></a>, <a class="txt" href="#E1:22_4_5_1"><span class="lab"><font color="Green" title="E59">A326</font></span></a></span>;<br/>
<span class="kw">assume </span><a NAME="E6:22_4_5_1"/>
<font color="Maroon" title="c25">m0</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c26">n</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"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c19">xD</font> <a href="normsp_0.html#K1" title="NORMSP_0:func.1">.</a> <font color="Maroon" title="c26">n</font></span>)</span> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <font color="Maroon" title="c22">v0</font></span>)</span></span><a href="normsp_0.html#K2" title="NORMSP_0:func.2">.||</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c23">r</font></span><br/>

<a NAME="E7:22_4_5_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"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="integra3.html#K2" title="INTEGRA3:func.2">delta</a> <font color="Maroon" title="c16">T</font></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c26">n</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span>)</span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c24">s</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:22_4_5"><span class="lab"><font color="Green" title="E57">A324</font></span></a></span>;<br/>
<a NAME="E8:22_4_5_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="integra3.html#K1" title="INTEGRA3:func.1">delta</a> <span class="p3">(<span class="default"><font color="Maroon" title="c16">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c26">n</font></span>)</span></span>)</span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c24">s</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:22_4_5_1"><span class="lab"><font color="Green" title="E59">A326</font></span></a>, <a class="ref" href="integra3.html#D2" onmouseover="rs('integra3/D2')" onmouseout="rh()">INTEGRA3:def 2</a></span>;<br/>
<a NAME="E9:22_4_5_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E62">A329</font></span>: 
 <a href="integra3.html#K1" title="INTEGRA3:func.1">delta</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c26">n</font></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c24">s</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="absvalue.html#D1" onmouseover="rs('absvalue/D1')" onmouseout="rh()">ABSVALUE:def 1</a>, <a class="ref" href="integra3.html#T9" onmouseover="rs('integra3/T9')" onmouseout="rh()">INTEGRA3:9</a></span>;<br/>
<a NAME="E10:22_4_5_1"/>
<font color="Maroon" title="c19">xD</font> <a href="normsp_0.html#K1" title="NORMSP_0:func.1">.</a> <font color="Maroon" title="c26">n</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="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>
 
;<br/>
<a NAME="E11:22_4_5_1"/><span class="kw">then </span>
<font color="Maroon" title="c19">xD</font> <a href="normsp_0.html#K1" title="NORMSP_0:func.1">.</a> <font color="Maroon" title="c26">n</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="c0sp1.html#K6" title="C0SP1:func.6">BoundedFunctions</a> <font color="Maroon" title="c1">A</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="dualsp05.html#T5" target="_self" onmouseover="rs('dualsp05/T5')" onmouseout="rh()">Lm1</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c29">g1</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><span class="kw"> such that </span><br/><a NAME="E13:22_4_5_1"/><span class="lab"><font color="Green" title="E63">A330</font></span>: 
( <font color="Maroon" title="c29">g1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">xD</font> <a href="normsp_0.html#K1" title="NORMSP_0:func.1">.</a> <font color="Maroon" title="c26">n</font> &amp; <font color="Maroon" title="c29">g1</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c1">A</font> is  <a href="comseq_2.html#V1" title="COMSEQ_2:attr.1">bounded</a>  )
 ;<br/>
<a NAME="E14:22_4_5_1"/><span class="lab"><font color="Green" title="E64">A331</font></span>: 
 for <font color="Olive" title="b1">t</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c1">A</font><br/>  for <font color="Olive" title="b2">i</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Olive" title="b2">i</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="c16">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c26">n</font></span>)</span> &amp; <font color="Olive" title="b1">t</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c16">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c26">n</font></span>)</span>,<font color="Olive" title="b2">i</font>) &amp; ( <font color="Olive" title="b2">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or (  <a href="seq_4.html#K3" title="SEQ_4:func.3">lower_bound</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c16">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c26">n</font></span>)</span>,<font color="Olive" title="b2">i</font>)</span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">t</font> &amp; <font color="Olive" title="b1">t</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="seq_4.html#K2" title="SEQ_4:func.2">upper_bound</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c16">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c26">n</font></span>)</span>,<font color="Olive" title="b2">i</font>)</span>)</span> ) ) holds <br/><font color="Maroon" title="c29">g1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c27">p2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">i</font></span>)</span>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="22_4_5_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c30">t</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c1">A</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">  for <font color="Olive" title="b1">i</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Olive" title="b1">i</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="c16">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c26">n</font></span>)</span> &amp; <font color="Maroon" title="c30">t</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c16">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c26">n</font></span>)</span>,<font color="Olive" title="b1">i</font>) &amp; ( <font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or (  <a href="seq_4.html#K3" title="SEQ_4:func.3">lower_bound</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c16">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c26">n</font></span>)</span>,<font color="Olive" title="b1">i</font>)</span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c30">t</font> &amp; <font color="Maroon" title="c30">t</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="seq_4.html#K2" title="SEQ_4:func.2">upper_bound</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c16">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c26">n</font></span>)</span>,<font color="Olive" title="b1">i</font>)</span>)</span> ) ) holds <br/><font color="Maroon" title="c29">g1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c30">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c27">p2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font></span>)</span></span><br/><span class="kw">let </span><font color="Maroon" title="c31">i</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="c31">i</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="c16">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c26">n</font></span>)</span> &amp; <font color="Maroon" title="c30">t</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c16">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c26">n</font></span>)</span>,<font color="Maroon" title="c31">i</font>) &amp; ( <font color="Maroon" title="c31">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or (  <a href="seq_4.html#K3" title="SEQ_4:func.3">lower_bound</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c16">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c26">n</font></span>)</span>,<font color="Maroon" title="c31">i</font>)</span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c30">t</font> &amp; <font color="Maroon" title="c30">t</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="seq_4.html#K2" title="SEQ_4:func.2">upper_bound</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c16">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c26">n</font></span>)</span>,<font color="Maroon" title="c31">i</font>)</span>)</span> ) ) implies <font color="Maroon" title="c29">g1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c30">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c27">p2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c31">i</font></span>)</span> )</span><br/>



<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:22_4_5_1_1"/><span class="lab"><font color="Green" title="E65">A332</font></span>: 
<font color="Maroon" title="c31">i</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="c16">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c26">n</font></span>)</span>
 <span class="kw">and </span><br/><a NAME="E2:22_4_5_1_1"/><span class="lab"><font color="Green" title="E66">A333</font></span>: 
<font color="Maroon" title="c30">t</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c16">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c26">n</font></span>)</span>,<font color="Maroon" title="c31">i</font>)
 <span class="kw">and </span><br/><a NAME="E3:22_4_5_1_1"/><span class="lab"><font color="Green" title="E67">A334</font></span>: 
( <font color="Maroon" title="c31">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or (  <a href="seq_4.html#K3" title="SEQ_4:func.3">lower_bound</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c16">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c26">n</font></span>)</span>,<font color="Maroon" title="c31">i</font>)</span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c30">t</font> &amp; <font color="Maroon" title="c30">t</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="seq_4.html#K2" title="SEQ_4:func.2">upper_bound</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c16">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c26">n</font></span>)</span>,<font color="Maroon" title="c31">i</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"> <font color="Maroon" title="c29">g1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c30">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c27">p2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c31">i</font></span>)</span></span><br/>

<span class="kw">consider </span><font color="Maroon" title="c32">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="E5:22_4_5_1_1"/><span class="lab"><font color="Green" title="E68">A336</font></span>: 
<font color="Maroon" title="c27">p2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c31">i</font> <a href="tarski.html#R2" title="TARSKI: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="c15">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="c16">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c26">n</font></span>)</span>,<font color="Maroon" title="c31">i</font>)</span>)</span></span>)</span>
 <span class="kw">and </span><br/><a NAME="E6:22_4_5_1_1"/><span class="lab"><font color="Green" title="E69">A337</font></span>: 
<font color="Maroon" title="c32">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">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="c16">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c26">n</font></span>)</span>,<font color="Maroon" title="c31">i</font>)</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c27">p2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c31">i</font></span>)</span>
 <span class="kw">and </span><br/><a NAME="E7:22_4_5_1_1"/><span class="lab"><font color="Green" title="E70">A338</font></span>: 
<font color="Maroon" title="c28">q2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c31">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c32">r</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="dualsp05.html#K2" title="DUALSP05:func.2">Dp1</a> (<font color="Maroon" title="c11">m</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c16">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c26">n</font></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon" title="c31">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>)</span>)</span> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <span class="p2">(<span class="default"><a href="dualsp05.html#K2" title="DUALSP05:func.2">Dp1</a> (<font color="Maroon" title="c11">m</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c16">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c26">n</font></span>)</span>,<font color="Maroon" title="c31">i</font>)</span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:22_4_5_1"><span class="lab"><font color="Green" title="E60">A327</font></span></a>, <a class="txt" href="#E5:22_4_5_1"><span class="lab"><font color="Green" title="E61">A328</font></span></a>, <a class="txt" href="#E1:22_4_5_1_1"><span class="lab"><font color="Green" title="E65">A332</font></span></a></span>;<br/>
<span class="kw">defpred </span><font color="Maroon">S<sub>3</sub></font>[  <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>,   <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a> ] <span class="kw">means </span> ex <font color="Olive" title="b1">qi</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> st <br/>( <font color="Olive" title="b1">qi</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c28">q2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> $1 &amp; $2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">qi</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c30">t</font> );<br/>
<a NAME="E8:22_4_5_1_1"/><span class="lab"><font color="Green" title="E71">A340</font></span>: 
 for <font color="Olive" title="b1">i</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Olive" title="b1">i</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="c16">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c26">n</font></span>)</span></span>)</span> holds <br/> ex <font color="Olive" title="b2">x</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  st <font color="Maroon">S<sub>3</sub></font>[<font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">x</font>]
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="22_4_5_1_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c33">i</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="c33">i</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="c16">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c26">n</font></span>)</span></span>)</span> implies  ex <font color="Olive" title="b1">x</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  st <font color="Maroon">S<sub>3</sub></font>[<font color="Maroon" title="c33">i</font>,<font color="Olive" title="b1">x</font>] )</span><br/>

<span class="kw">assume </span><a NAME="E1:22_4_5_1_1_1"/>
<font color="Maroon" title="c33">i</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="c16">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c26">n</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">  ex <font color="Olive" title="b1">x</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  st <font color="Maroon">S<sub>3</sub></font>[<font color="Maroon" title="c33">i</font>,<font color="Olive" title="b1">x</font>]</span><br/>

<a NAME="E2:22_4_5_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c33">i</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="c16">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c26">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">then </span><span class="kw">consider </span><font color="Maroon" title="c34">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="E4:22_4_5_1_1_1"/><span class="lab"><font color="Green" title="E72">A342</font></span>: 
( <font color="Maroon" title="c27">p2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c33">i</font> <a href="tarski.html#R2" title="TARSKI: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="c15">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="c16">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c26">n</font></span>)</span>,<font color="Maroon" title="c33">i</font>)</span>)</span></span>)</span> &amp; <font color="Maroon" title="c34">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">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="c16">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c26">n</font></span>)</span>,<font color="Maroon" title="c33">i</font>)</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c27">p2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c33">i</font></span>)</span> &amp; <font color="Maroon" title="c28">q2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c33">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c34">r</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="dualsp05.html#K2" title="DUALSP05:func.2">Dp1</a> (<font color="Maroon" title="c11">m</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c16">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c26">n</font></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon" title="c33">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>)</span>)</span> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <span class="p2">(<span class="default"><a href="dualsp05.html#K2" title="DUALSP05:func.2">Dp1</a> (<font color="Maroon" title="c11">m</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c16">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c26">n</font></span>)</span>,<font color="Maroon" title="c33">i</font>)</span>)</span></span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:22_4_5_1"><span class="lab"><font color="Green" title="E60">A327</font></span></a>, <a class="txt" href="#E5:22_4_5_1"><span class="lab"><font color="Green" title="E61">A328</font></span></a></span>;<br/>
<a NAME="E5:22_4_5_1_1_1"/>
<font color="Maroon" title="c28">q2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c33">i</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="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="txt" href="#E4:22_4_5_1_1_1"><span class="lab"><font color="Green" title="E72">A342</font></span></a></span>;<br/>
<a NAME="E6:22_4_5_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c28">q2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c33">i</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="c0sp1.html#K6" title="C0SP1:func.6">BoundedFunctions</a> <font color="Maroon" title="c1">A</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="dualsp05.html#T5" target="_self" onmouseover="rs('dualsp05/T5')" onmouseout="rh()">Lm1</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c35">qi</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><span class="kw"> such that </span><br/><a NAME="E8:22_4_5_1_1_1"/><span class="lab"><font color="Green" title="E73">A343</font></span>: 
( <font color="Maroon" title="c35">qi</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c28">q2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c33">i</font> &amp; <font color="Maroon" title="c35">qi</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c1">A</font> is  <a href="comseq_2.html#V1" title="COMSEQ_2:attr.1">bounded</a>  )
 ;<br/>
<span class="kw">take </span><font color="Maroon" title="c36">x</font> = <font color="Maroon" title="c35">qi</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c30">t</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">S<sub>3</sub></font>[<font color="Maroon" title="c33">i</font>,<font color="Maroon" title="c36">x</font>]</span><br/>

<span class="kw">thus </span><a NAME="E9:22_4_5_1_1_1"/>
<font color="Maroon">S<sub>3</sub></font>[<font color="Maroon" title="c33">i</font>,<font color="Maroon" title="c36">x</font>]
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:22_4_5_1_1_1"><span class="lab"><font color="Green" title="E73">A343</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">consider </span><font color="Maroon" title="c33">z</font> being    <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> <span class="kw"> such that </span><br/><a NAME="E10:22_4_5_1_1"/><span class="lab"><font color="Green" title="E72">A344</font></span>: 
(  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c33">z</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> <span class="p2">(<span class="default"><font color="Maroon" title="c16">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c26">n</font></span>)</span></span>)</span> &amp; (  for <font color="Olive" title="b1">i</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Olive" title="b1">i</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="c16">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c26">n</font></span>)</span></span>)</span> holds <br/><font color="Maroon">S<sub>3</sub></font>[<font color="Olive" title="b1">i</font>,<font color="Maroon" title="c33">z</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font>] ) )
 <span class="kw">from</span> <span class="lab"><a class="ref" href="finseq_1.html#S5" onmouseover="rs('finseq_1/S5')" onmouseout="rh()">FINSEQ_1:sch 5</a>(<a class="txt" href="#E8:22_4_5_1_1"><span class="lab"><font color="Green" title="E71">A340</font></span></a>);<br/></span>
<a NAME="E11:22_4_5_1_1"/><span class="lab"><font color="Green" title="E73">A346</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c28">q2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c33">z</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:22_4_5_1"><span class="lab"><font color="Green" title="E61">A328</font></span></a>, <a class="txt" href="#E10:22_4_5_1_1"><span class="lab"><font color="Green" title="E72">A344</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="E12:22_4_5_1_1"/><span class="lab"><font color="Green" title="E74">A347</font></span>: 
<font color="Maroon" title="c31">i</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> <font color="Maroon" title="c33">z</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:22_4_5_1_1"><span class="lab"><font color="Green" title="E72">A344</font></span></a>, <a class="txt" href="#E1:22_4_5_1_1"><span class="lab"><font color="Green" title="E65">A332</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="E13:22_4_5_1_1"/><span class="lab"><font color="Green" title="E75">A348</font></span>: 
<font color="Maroon" title="c29">g1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c30">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <font color="Maroon" title="c33">z</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:22_4_5_1"><span class="lab"><font color="Green" title="E61">A328</font></span></a>, <a class="txt" href="#E13:22_4_5_1"><span class="lab"><font color="Green" title="E63">A330</font></span></a>, <a class="txt" href="#E11:22_4_5_1_1"><span class="lab"><font color="Green" title="E73">A346</font></span></a>, <a class="ref" href="dualsp05.html#T12" target="_self" onmouseover="rs('dualsp05/T12')" onmouseout="rh()">LM89</a>, <a class="txt" href="#E10:22_4_5_1_1"><span class="lab"><font color="Green" title="E72">A344</font></span></a></span>;<br/>
<a NAME="E14:22_4_5_1_1"/><span class="lab"><font color="Green" title="E76">A349</font></span>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c33">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c26">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>, <a class="txt" href="#E10:22_4_5_1_1"><span class="lab"><font color="Green" title="E72">A344</font></span></a></span>;<br/>
<a NAME="E15:22_4_5_1_1"/><span class="lab"><font color="Green" title="E77">A350</font></span>: 
<font color="Maroon" title="c31">i</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="c16">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c26">n</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:22_4_5_1_1"><span class="lab"><font color="Green" title="E65">A332</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/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c34">qi</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><span class="kw"> such that </span><br/><a NAME="E17:22_4_5_1_1"/><span class="lab"><font color="Green" title="E78">A351</font></span>: 
( <font color="Maroon" title="c34">qi</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c28">q2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c31">i</font> &amp; <font color="Maroon" title="c33">z</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c31">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c34">qi</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c30">t</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:22_4_5_1_1"><span class="lab"><font color="Green" title="E72">A344</font></span></a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c35">D</font> = <font color="Maroon" title="c16">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c26">n</font>;<br/>
<a NAME="E18:22_4_5_1_1"/><span class="lab"><font color="Green" title="E79">B352</font></span>: 
<font color="Maroon" title="c30">t</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><span class="p2">(<span class="default"><a href="seq_4.html#K3" title="SEQ_4:func.3">lower_bound</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="c16">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c26">n</font></span>)</span>,<font color="Maroon" title="c31">i</font>)</span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="seq_4.html#K2" title="SEQ_4:func.2">upper_bound</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="c16">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c26">n</font></span>)</span>,<font color="Maroon" title="c31">i</font>)</span>)</span></span>)</span></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:22_4_5_1_1"><span class="lab"><font color="Green" title="E66">A333</font></span></a>, <a class="ref" href="integra1.html#T4" onmouseover="rs('integra1/T4')" onmouseout="rh()">INTEGRA1:4</a></span>;<br/>
<a NAME="E19:22_4_5_1_1"/><span class="lab"><font color="Green" title="E80">A354</font></span>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c31">i</font> &amp; <font color="Maroon" title="c31">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c26">n</font></span>)</span> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:22_4_5_1_1"><span class="lab"><font color="Green" title="E77">A350</font></span></a>, <a class="ref" href="finseq_1.html#T1" onmouseover="rs('finseq_1/T1')" onmouseout="rh()">FINSEQ_1:1</a></span>;<br/>
<a NAME="E20:22_4_5_1_1"/><span class="kw">then </span>
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c31">i</font> &amp; <font color="Maroon" title="c31">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</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="c16">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c26">n</font></span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T7" onmouseover="rs('xreal_1/T7')" onmouseout="rh()">XREAL_1:7</a></span>;<br/>
<a NAME="E21:22_4_5_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E81">A355</font></span>: 
<font color="Maroon" title="c31">i</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"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c16">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c26">n</font></span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
 
;<br/>
<a NAME="E22:22_4_5_1_1"/>
( 1 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c31">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <font color="Maroon" title="c31">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c26">n</font></span>)</span> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:22_4_5_1_1"><span class="lab"><font color="Green" title="E77">A350</font></span></a>, <a class="ref" href="finseq_1.html#T1" onmouseover="rs('finseq_1/T1')" onmouseout="rh()">FINSEQ_1:1</a>, <a class="ref" href="xreal_1.html#T7" onmouseover="rs('xreal_1/T7')" onmouseout="rh()">XREAL_1:7</a></span>;<br/>
<a NAME="E23:22_4_5_1_1"/><span class="kw">then </span>
( 1 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c31">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <font color="Maroon" title="c31">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</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="c16">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c26">n</font></span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T7" onmouseover="rs('xreal_1/T7')" onmouseout="rh()">XREAL_1:7</a></span>;<br/>
<a NAME="E24:22_4_5_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E82">A356</font></span>: 
<font color="Maroon" title="c31">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <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"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c16">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c26">n</font></span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
 
;<br/>
<a NAME="E25:22_4_5_1_1"/><span class="lab"><font color="Green" title="E83">A357</font></span>: 
<font color="Maroon" title="c33">z</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c31">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c32">r</font>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="22_4_5_1_1_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">set </span><font color="Maroon" title="c36">f0</font> = <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c9">a</font>,<font color="Maroon" title="c10">b</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>;<br/>
<span class="kw">set </span><font color="Maroon" title="c37">f1</font> = <span class="p1">(<span class="default"><span class="p2"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c9">a</font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c16">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c26">n</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c31">i</font></span>)</span></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> 1</span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p1">(<span class="default"><span class="p2"><a href="rcomp_1.html#K4" title="RCOMP_1:func.4">].</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c16">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c26">n</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c31">i</font></span>)</span>,<font color="Maroon" title="c10">b</font></span><a href="rcomp_1.html#K4" title="RCOMP_1:func.4">.]</a></span> <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span>)</span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c38">g1</font> = <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c9">a</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c16">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c26">n</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c31">i</font></span>)</span></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> 1;<br/>
<span class="kw">set </span><font color="Maroon" title="c39">g2</font> = <span class="p1"><a href="rcomp_1.html#K4" title="RCOMP_1:func.4">].</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c16">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c26">n</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c31">i</font></span>)</span>,<font color="Maroon" title="c10">b</font></span><a href="rcomp_1.html#K4" title="RCOMP_1:func.4">.]</a></span> <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>;<br/>
<span class="kw">set </span><font color="Maroon" title="c40">f2</font> = <span class="p1">(<span class="default"><span class="p2"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c9">a</font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c16">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c26">n</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c31">i</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span></span>)</span></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> 1</span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p1">(<span class="default"><span class="p2"><a href="rcomp_1.html#K4" title="RCOMP_1:func.4">].</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c16">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c26">n</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c31">i</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c10">b</font></span><a href="rcomp_1.html#K4" title="RCOMP_1:func.4">.]</a></span> <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span>)</span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c41">h1</font> = <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c9">a</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c16">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c26">n</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c31">i</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span></span>)</span></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> 1;<br/>
<span class="kw">set </span><font color="Maroon" title="c42">h2</font> = <span class="p1"><a href="rcomp_1.html#K4" title="RCOMP_1:func.4">].</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c16">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c26">n</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c31">i</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c10">b</font></span><a href="rcomp_1.html#K4" title="RCOMP_1:func.4">.]</a></span> <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>;<br/>
<a NAME="E1:22_4_5_1_1_2"/><span class="lab"><font color="Green" title="E84">B20</font></span>: 
(  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c9">a</font>,<font color="Maroon" title="c10">b</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c9">a</font>,<font color="Maroon" title="c10">b</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> &amp;  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c9">a</font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c16">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c26">n</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c31">i</font></span>)</span></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c9">a</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c16">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c26">n</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c31">i</font></span>)</span></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> &amp;  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2"><a href="rcomp_1.html#K4" title="RCOMP_1:func.4">].</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c16">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c26">n</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c31">i</font></span>)</span>,<font color="Maroon" title="c10">b</font></span><a href="rcomp_1.html#K4" title="RCOMP_1:func.4">.]</a></span> <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="rcomp_1.html#K4" title="RCOMP_1:func.4">].</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c16">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c26">n</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c31">i</font></span>)</span>,<font color="Maroon" title="c10">b</font></span><a href="rcomp_1.html#K4" title="RCOMP_1:func.4">.]</a></span> &amp;  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c9">a</font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c16">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c26">n</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c31">i</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span></span>)</span></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c9">a</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c16">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c26">n</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c31">i</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span></span>)</span></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> &amp;  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2"><a href="rcomp_1.html#K4" title="RCOMP_1:func.4">].</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c16">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c26">n</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c31">i</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c10">b</font></span><a href="rcomp_1.html#K4" title="RCOMP_1:func.4">.]</a></span> <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="rcomp_1.html#K4" title="RCOMP_1:func.4">].</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c16">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c26">n</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c31">i</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c10">b</font></span><a href="rcomp_1.html#K4" title="RCOMP_1:func.4">.]</a></span> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funcop_1.html#T13" onmouseover="rs('funcop_1/T13')" onmouseout="rh()">FUNCOP_1:13</a></span>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="22_4_5_1_1_2_1"><span class="kw">per </span></a><a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:22_4_5_1_1_2_1"/>
( <font color="Maroon" title="c31">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or <font color="Maroon" title="c31">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 1 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="22_4_5_1_1_2_1_1"><span class="kw">suppose </span></a><a NAME="E1:22_4_5_1_1_2_1_1"/><span class="lab"><font color="Green" title="E85">A358</font></span>: 
<font color="Maroon" title="c31">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 ; <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="c33">z</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c31">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c32">r</font></span><br/><div class="add"><a NAME="E2:22_4_5_1_1_2_1_1"/>
<font color="Maroon" title="c1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><span class="p2">(<span class="default"><a href="seq_4.html#K3" title="SEQ_4:func.3">lower_bound</a> <font color="Maroon" title="c1">A</font></span>)</span>,<span class="p2">(<span class="default"><a href="seq_4.html#K2" title="SEQ_4:func.2">upper_bound</a> <font color="Maroon" title="c1">A</font></span>)</span></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="integra1.html#T4" onmouseover="rs('integra1/T4')" onmouseout="rh()">INTEGRA1:4</a></span>;<br/><a NAME="E3:22_4_5_1_1_2_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E86">A359</font></span>: 
 <a href="seq_4.html#K3" title="SEQ_4:func.3">lower_bound</a> <font color="Maroon" title="c1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">a</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:22"><span class="lab"><font color="Green" title="E28">A7</font></span></a>, <a class="ref" href="integra1.html#T5" onmouseover="rs('integra1/T5')" onmouseout="rh()">INTEGRA1:5</a></span>;<br/><a NAME="E4:22_4_5_1_1_2_1_1"/><span class="lab"><font color="Green" title="E87">A360</font></span>: 
<font color="Maroon" title="c9">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c9">a</font>,<font color="Maroon" title="c10">b</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:22"><span class="lab"><font color="Green" title="E28">A7</font></span></a></span>;<br/><a NAME="E5:22_4_5_1_1_2_1_1"/><span class="lab"><font color="Green" title="E88">A361</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c16">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c26">n</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c31">i</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c9">a</font>,<font color="Maroon" title="c10">b</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:22"><span class="lab"><font color="Green" title="E28">A7</font></span></a>, <a class="txt" href="#E1:22_4_5_1_1"><span class="lab"><font color="Green" title="E65">A332</font></span></a>, <a class="ref" href="integra1.html#T6" onmouseover="rs('integra1/T6')" onmouseout="rh()">INTEGRA1:6</a></span>;<br/><a NAME="E6:22_4_5_1_1_2_1_1"/><span class="lab"><font color="Green" title="E89">A362</font></span>: 
(  <a href="seq_4.html#K3" title="SEQ_4:func.3">lower_bound</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c16">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c26">n</font></span>)</span>,<font color="Maroon" title="c31">i</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="seq_4.html#K3" title="SEQ_4:func.3">lower_bound</a> <font color="Maroon" title="c1">A</font> &amp;  <a href="seq_4.html#K2" title="SEQ_4:func.2">upper_bound</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c16">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c26">n</font></span>)</span>,<font color="Maroon" title="c31">i</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c26">n</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c31">i</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:22_4_5_1_1"><span class="lab"><font color="Green" title="E65">A332</font></span></a>, <a class="txt" href="#E1:22_4_5_1_1_2_1_1"><span class="lab"><font color="Green" title="E85">A358</font></span></a>, <a class="ref" href="integra1.html#D4" onmouseover="rs('integra1/D4')" onmouseout="rh()">INTEGRA1:def 4</a></span>;<br/><a NAME="E7:22_4_5_1_1_2_1_1"/><span class="lab"><font color="Green" title="E90">A364</font></span>: 
<font color="Maroon" title="c9">a</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c26">n</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c31">i</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:22_4"><span class="lab"><font color="Green" title="E42">A264</font></span></a>, <a class="txt" href="#E1:22_4_5_1_1_2_1_1"><span class="lab"><font color="Green" title="E85">A358</font></span></a></span>;<br/><span class="lab"><font color="Green" title="E91">A365</font></span>:  <a href="dualsp05.html#K2" title="DUALSP05:func.2">Dp1</a> (<font color="Maroon" title="c11">m</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c16">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c26">n</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c31">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>) = 
<font color="Maroon" title="c11">m</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c16">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c26">n</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c31">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E24:22_4_5_1_1"><span class="lab"><font color="Green" title="E82">A356</font></span></a>, <a class="ref" href="dualsp05.html#D2" target="_self" onmouseover="rs('dualsp05/D2')" onmouseout="rh()">defDp1</a>, <a class="txt" href="#E1:22_4_5_1_1_2_1_1"><span class="lab"><font color="Green" title="E85">A358</font></span></a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c9">a</font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c16">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c26">n</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c31">i</font></span>)</span></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> 1</span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p1">(<span class="default"><span class="p2"><a href="rcomp_1.html#K4" title="RCOMP_1:func.4">].</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c16">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c26">n</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c31">i</font></span>)</span>,<font color="Maroon" title="c10">b</font></span><a href="rcomp_1.html#K4" title="RCOMP_1:func.4">.]</a></span> <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:22"><span class="lab"><font color="Green" title="E29">A8</font></span></a>, <a class="txt" href="#E5:22_4_5_1_1_2_1_1"><span class="lab"><font color="Green" title="E88">A361</font></span></a>, <a class="txt" href="#E7:22_4_5_1_1_2_1_1"><span class="lab"><font color="Green" title="E90">A364</font></span></a></span>
;<br/><span class="lab"><font color="Green" title="E92">A366</font></span>:  <a href="dualsp05.html#K2" title="DUALSP05:func.2">Dp1</a> (<font color="Maroon" title="c11">m</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c16">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c26">n</font></span>)</span>,<font color="Maroon" title="c31">i</font>) = 
<font color="Maroon" title="c11">m</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="seq_4.html#K3" title="SEQ_4:func.3">lower_bound</a> <font color="Maroon" title="c1">A</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E21:22_4_5_1_1"><span class="lab"><font color="Green" title="E81">A355</font></span></a>, <a class="txt" href="#E1:22_4_5_1_1_2_1_1"><span class="lab"><font color="Green" title="E85">A358</font></span></a>, <a class="ref" href="dualsp05.html#D2" target="_self" onmouseover="rs('dualsp05/D2')" onmouseout="rh()">defDp1</a></span>
<br/>.= 
<span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c9">a</font>,<font color="Maroon" title="c10">b</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:22"><span class="lab"><font color="Green" title="E29">A8</font></span></a>, <a class="txt" href="#E3:22_4_5_1_1_2_1_1"><span class="lab"><font color="Green" title="E86">A359</font></span></a>, <a class="txt" href="#E4:22_4_5_1_1_2_1_1"><span class="lab"><font color="Green" title="E87">A360</font></span></a></span>
;<br/><a NAME="E10:22_4_5_1_1_2_1_1"/><span class="lab"><font color="Green" title="E93">A368</font></span>: 
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c9">a</font>,<font color="Maroon" title="c10">b</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">A</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:22"><span class="lab"><font color="Green" title="E28">A7</font></span></a>, <a class="ref" href="funcop_1.html#T13" onmouseover="rs('funcop_1/T13')" onmouseout="rh()">FUNCOP_1:13</a></span>;<br/><a NAME="E11:22_4_5_1_1_2_1_1"/><span class="lab"><font color="Green" title="E94">A369</font></span>: 
( <font color="Maroon" title="c9">a</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c26">n</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c31">i</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c16">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c26">n</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c31">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c10">b</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:22_4_5_1_1_2_1_1"><span class="lab"><font color="Green" title="E88">A361</font></span></a>, <a class="ref" href="xxreal_1.html#T1" onmouseover="rs('xxreal_1/T1')" onmouseout="rh()">XXREAL_1:1</a></span>;<br/><span class="lab"><font color="Green" title="E95">A370</font></span>:  <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c9">a</font>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c16">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c26">n</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c31">i</font></span>)</span></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> 1</span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p2">(<span class="default"><span class="p3"><a href="rcomp_1.html#K4" title="RCOMP_1:func.4">].</a><span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c16">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c26">n</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c31">i</font></span>)</span>,<font color="Maroon" title="c10">b</font></span><a href="rcomp_1.html#K4" title="RCOMP_1:func.4">.]</a></span> <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p2">(<span class="default"><span class="p3"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c9">a</font>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c16">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c26">n</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c31">i</font></span>)</span></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> 1</span>)</span></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p2">(<span class="default"><span class="p3"><a href="rcomp_1.html#K4" title="RCOMP_1:func.4">].</a><span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c16">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c26">n</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c31">i</font></span>)</span>,<font color="Maroon" title="c10">b</font></span><a href="rcomp_1.html#K4" title="RCOMP_1:func.4">.]</a></span> <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_4.html#D1" onmouseover="rs('funct_4/D1')" onmouseout="rh()">FUNCT_4:def 1</a></span>
<br/>.= 
<font color="Maroon" title="c1">A</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:22"><span class="lab"><font color="Green" title="E28">A7</font></span></a>, <a class="txt" href="#E1:22_4_5_1_1_2"><span class="lab"><font color="Green" title="E84">B20</font></span></a>, <a class="txt" href="#E11:22_4_5_1_1_2_1_1"><span class="lab"><font color="Green" title="E94">A369</font></span></a>, <a class="ref" href="xxreal_1.html#T167" onmouseover="rs('xxreal_1/T167')" onmouseout="rh()">XXREAL_1:167</a></span>
;<br/><a NAME="E13:22_4_5_1_1_2_1_1"/>
 <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <span class="p1">(<span class="default"><span class="p2"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c9">a</font>,<font color="Maroon" title="c10">b</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> 
 ;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c43">f0</font> = <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c9">a</font>,<font color="Maroon" title="c10">b</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> as   <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> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:22_4_5_1_1_2_1_1"><span class="lab"><font color="Green" title="E93">A368</font></span></a>, <a class="ref" href="funct_2.html#T2" onmouseover="rs('funct_2/T2')" onmouseout="rh()">FUNCT_2:2</a></span>;<br/><a NAME="E15:22_4_5_1_1_2_1_1"/>
 <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c9">a</font>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c16">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c26">n</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c31">i</font></span>)</span></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> 1</span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p2">(<span class="default"><span class="p3"><a href="rcomp_1.html#K4" title="RCOMP_1:func.4">].</a><span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c16">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c26">n</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c31">i</font></span>)</span>,<font color="Maroon" title="c10">b</font></span><a href="rcomp_1.html#K4" title="RCOMP_1:func.4">.]</a></span> <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span>)</span></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> 
 ;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c44">f1</font> = <span class="p1">(<span class="default"><span class="p2"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c9">a</font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c16">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c26">n</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c31">i</font></span>)</span></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> 1</span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p1">(<span class="default"><span class="p2"><a href="rcomp_1.html#K4" title="RCOMP_1:func.4">].</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c16">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c26">n</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c31">i</font></span>)</span>,<font color="Maroon" title="c10">b</font></span><a href="rcomp_1.html#K4" title="RCOMP_1:func.4">.]</a></span> <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span>)</span> as   <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> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:22_4_5_1_1_2_1_1"><span class="lab"><font color="Green" title="E95">A370</font></span></a>, <a class="ref" href="funct_2.html#T2" onmouseover="rs('funct_2/T2')" onmouseout="rh()">FUNCT_2:2</a></span>;<br/><a NAME="E17:22_4_5_1_1_2_1_1"/><span class="lab"><font color="Green" title="E96">A372</font></span>: 
<font color="Maroon" title="c34">qi</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c30">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c32">r</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c44">f1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c30">t</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c43">f0</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c30">t</font></span>)</span></span>)</span>
 <div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="22_4_5_1_1_2_1_1_4"><span class="kw">proof </span></a><div class="add">

<span class="kw">reconsider </span><font color="Maroon" title="c45">H</font> = <span class="p1">(<span class="default"><a href="dualsp05.html#K2" title="DUALSP05:func.2">Dp1</a> (<font color="Maroon" title="c11">m</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c16">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c26">n</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c31">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>)</span>)</span> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <span class="p1">(<span class="default"><a href="dualsp05.html#K2" title="DUALSP05:func.2">Dp1</a> (<font color="Maroon" title="c11">m</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c16">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c26">n</font></span>)</span>,<font color="Maroon" title="c31">i</font>)</span>)</span> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="c0sp1.html#K11" title="C0SP1:func.11">R_Normed_Algebra_of_BoundedFunctions</a> <font color="Maroon" title="c1">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="c46">h</font> = <font color="Maroon" title="c45">H</font> as   <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> <span class="kw">by</span> <span class="lab"><a class="ref" href="dualsp05.html#T11" target="_self" onmouseover="rs('dualsp05/T11')" onmouseout="rh()">LM88</a></span>;<br/>
<a NAME="E3:22_4_5_1_1_2_1_1_4"/>
<font color="Maroon" title="c34">qi</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c32">r</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c45">H</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="dualsp05.html#T5" target="_self" onmouseover="rs('dualsp05/T5')" onmouseout="rh()">Lm1</a>, <a class="txt" href="#E17:22_4_5_1_1"><span class="lab"><font color="Green" title="E78">A351</font></span></a>, <a class="txt" href="#E7:22_4_5_1_1"><span class="lab"><font color="Green" title="E70">A338</font></span></a></span>;<br/>
<span class="kw">then </span><font color="Maroon" title="c34">qi</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c30">t</font> = 
<font color="Maroon" title="c32">r</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c46">h</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c30">t</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="c0sp1.html#T30" onmouseover="rs('c0sp1/T30')" onmouseout="rh()">C0SP1:30</a></span>
<br/>.= 
<font color="Maroon" title="c32">r</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c44">f1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c30">t</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c43">f0</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c30">t</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:22"><span class="lab"><font color="Green" title="E23">A2</font></span></a>, <a class="txt" href="#E8:22_4_5_1_1_2_1_1"><span class="lab"><font color="Green" title="E91">A365</font></span></a>, <a class="txt" href="#E9:22_4_5_1_1_2_1_1"><span class="lab"><font color="Green" title="E92">A366</font></span></a>, <a class="ref" href="c0sp1.html#T34" onmouseover="rs('c0sp1/T34')" onmouseout="rh()">C0SP1:34</a></span>
;<br/>
<span class="kw">hence </span><a NAME="E5:22_4_5_1_1_2_1_1_4"/>
<font color="Maroon" title="c34">qi</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c30">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c32">r</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c44">f1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c30">t</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c43">f0</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c30">t</font></span>)</span></span>)</span>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><a NAME="E18:22_4_5_1_1_2_1_1"/><span class="lab"><font color="Green" title="E97">A373</font></span>: 
<font color="Maroon" title="c30">t</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c9">a</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c16">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c26">n</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c31">i</font></span>)</span></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:22_4_5_1_1_2_1_1"><span class="lab"><font color="Green" title="E86">A359</font></span></a>, <a class="txt" href="#E6:22_4_5_1_1_2_1_1"><span class="lab"><font color="Green" title="E89">A362</font></span></a>, <a class="ref" href="integra1.html#T4" onmouseover="rs('integra1/T4')" onmouseout="rh()">INTEGRA1:4</a>, <a class="txt" href="#E2:22_4_5_1_1"><span class="lab"><font color="Green" title="E66">A333</font></span></a></span>;<br/><span class="kw">then </span><span class="lab"><font color="Green" title="E98">A375</font></span>: <font color="Maroon" title="c44">f1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c30">t</font> = 
<span class="p1">(<span class="default"><span class="p2"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c9">a</font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c16">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c26">n</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c31">i</font></span>)</span></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> 1</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c30">t</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_4.html#T16" onmouseover="rs('funct_4/T16')" onmouseout="rh()">FUNCT_4:16</a>, <a class="txt" href="#E1:22_4_5_1_1_2"><span class="lab"><font color="Green" title="E84">B20</font></span></a>, <a class="ref" href="xxreal_1.html#T90" onmouseover="rs('xxreal_1/T90')" onmouseout="rh()">XXREAL_1:90</a></span>
<br/>.= 
1
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:22_4_5_1_1_2_1_1"><span class="lab"><font color="Green" title="E97">A373</font></span></a>, <a class="ref" href="funcop_1.html#T7" onmouseover="rs('funcop_1/T7')" onmouseout="rh()">FUNCOP_1:7</a></span>
;<br/><a NAME="E20:22_4_5_1_1_2_1_1"/>
<font color="Maroon" title="c43">f0</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c30">t</font> <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="txt" href="#E11:22"><span class="lab"><font color="Green" title="E28">A7</font></span></a>, <a class="ref" href="funcop_1.html#T7" onmouseover="rs('funcop_1/T7')" onmouseout="rh()">FUNCOP_1:7</a></span>;<br/><span class="kw">hence </span><a NAME="E21:22_4_5_1_1_2_1_1"/>
<font color="Maroon" title="c33">z</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c31">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c32">r</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:22_4_5_1_1"><span class="lab"><font color="Green" title="E78">A351</font></span></a>, <a class="txt" href="#E17:22_4_5_1_1_2_1_1"><span class="lab"><font color="Green" title="E96">A372</font></span></a>, <a class="txt" href="#E19:22_4_5_1_1_2_1_1"><span class="lab"><font color="Green" title="E98">A375</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><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="22_4_5_1_1_2_1_2"><span class="kw">suppose </span></a><a NAME="E1:22_4_5_1_1_2_1_2"/><span class="lab"><font color="Green" title="E85">A376</font></span>: 
<font color="Maroon" title="c31">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 1
 ; <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="c33">z</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c31">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c32">r</font></span><br/><div class="add"><a NAME="E2:22_4_5_1_1_2_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E86">A378</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c16">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c26">n</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c31">i</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c9">a</font>,<font color="Maroon" title="c10">b</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:22"><span class="lab"><font color="Green" title="E28">A7</font></span></a>, <a class="txt" href="#E1:22_4_5_1_1"><span class="lab"><font color="Green" title="E65">A332</font></span></a>, <a class="ref" href="integra1.html#T7" onmouseover="rs('integra1/T7')" onmouseout="rh()">INTEGRA1:7</a></span>;<br/><a NAME="E3:22_4_5_1_1_2_1_2"/><span class="lab"><font color="Green" title="E87">A377</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c16">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c26">n</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c31">i</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c9">a</font>,<font color="Maroon" title="c10">b</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:22"><span class="lab"><font color="Green" title="E28">A7</font></span></a>, <a class="txt" href="#E1:22_4_5_1_1"><span class="lab"><font color="Green" title="E65">A332</font></span></a>, <a class="ref" href="integra1.html#T6" onmouseover="rs('integra1/T6')" onmouseout="rh()">INTEGRA1:6</a></span>;<br/><a NAME="E4:22_4_5_1_1_2_1_2"/><span class="lab"><font color="Green" title="E88">A379</font></span>: 
(  <a href="seq_4.html#K3" title="SEQ_4:func.3">lower_bound</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c16">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c26">n</font></span>)</span>,<font color="Maroon" title="c31">i</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c26">n</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c31">i</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span> &amp;  <a href="seq_4.html#K2" title="SEQ_4:func.2">upper_bound</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c16">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c26">n</font></span>)</span>,<font color="Maroon" title="c31">i</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c26">n</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c31">i</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:22_4_5_1_1"><span class="lab"><font color="Green" title="E65">A332</font></span></a>, <a class="txt" href="#E1:22_4_5_1_1_2_1_2"><span class="lab"><font color="Green" title="E85">A376</font></span></a>, <a class="ref" href="integra1.html#D4" onmouseover="rs('integra1/D4')" onmouseout="rh()">INTEGRA1:def 4</a></span>;<br/><a NAME="E5:22_4_5_1_1_2_1_2"/>
<font color="Maroon" title="c31">i</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 <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="c16">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c26">n</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:22_4_5_1_1"><span class="lab"><font color="Green" title="E65">A332</font></span></a>, <a class="txt" href="#E1:22_4_5_1_1_2_1_2"><span class="lab"><font color="Green" title="E85">A376</font></span></a>, <a class="ref" href="integra1.html#T7" onmouseover="rs('integra1/T7')" onmouseout="rh()">INTEGRA1:7</a></span>;<br/><a NAME="E6:22_4_5_1_1_2_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E89">A382</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c16">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c26">n</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c31">i</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c26">n</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c31">i</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:22_4_5_1_1"><span class="lab"><font color="Green" title="E65">A332</font></span></a>, <a class="ref" href="xreal_1.html#T146" onmouseover="rs('xreal_1/T146')" onmouseout="rh()">XREAL_1:146</a>, <a class="ref" href="valued_0.html#D13" onmouseover="rs('valued_0/D13')" onmouseout="rh()">VALUED_0:def 13</a></span>;<br/><a NAME="E7:22_4_5_1_1_2_1_2"/>
( <span class="p1">(<span class="default"><font color="Maroon" title="c16">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c26">n</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c31">i</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">a</font> or <span class="p1">(<span class="default"><font color="Maroon" title="c16">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c26">n</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c31">i</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"><a href="rcomp_1.html#K4" title="RCOMP_1:func.4">].</a><span class="default"><font color="Maroon" title="c9">a</font>,<font color="Maroon" title="c10">b</font></span><a href="rcomp_1.html#K4" title="RCOMP_1:func.4">.]</a></span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:22_4_5_1_1_2_1_2"><span class="lab"><font color="Green" title="E85">A376</font></span></a>, <a class="txt" href="#E11:22"><span class="lab"><font color="Green" title="E28">A7</font></span></a>, <a class="txt" href="#E1:22_4_5_1_1"><span class="lab"><font color="Green" title="E65">A332</font></span></a>, <a class="ref" href="integra1.html#T7" onmouseover="rs('integra1/T7')" onmouseout="rh()">INTEGRA1:7</a>, <a class="ref" href="xxreal_1.html#T6" onmouseover="rs('xxreal_1/T6')" onmouseout="rh()">XXREAL_1:6</a></span>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="22_4_5_1_1_2_1_2_1"><span class="kw">per </span></a><a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:22_4_5_1_1_2_1_2_1"/><span class="kw">then </span>
( <font color="Maroon" title="c9">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c26">n</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c31">i</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span> or <font color="Maroon" title="c9">a</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c26">n</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c31">i</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span> )
 </span><span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_1.html#T2" onmouseover="rs('xxreal_1/T2')" onmouseout="rh()">XXREAL_1:2</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="22_4_5_1_1_2_1_2_1_1"><span class="kw">suppose </span></a><a NAME="E1:22_4_5_1_1_2_1_2_1_1"/><span class="lab"><font color="Green" title="E90">A384</font></span>: 
<font color="Maroon" title="c9">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c26">n</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c31">i</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</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="c33">z</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c31">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c32">r</font></span><br/><div class="add"><a NAME="E2:22_4_5_1_1_2_1_2_1_1"/>
1 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c31">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T8" onmouseover="rs('xreal_1/T8')" onmouseout="rh()">XREAL_1:8</a>, <a class="txt" href="#E19:22_4_5_1_1"><span class="lab"><font color="Green" title="E80">A354</font></span></a></span>;<br/><span class="kw">then </span><span class="lab"><font color="Green" title="E91">A386</font></span>:  <a href="dualsp05.html#K2" title="DUALSP05:func.2">Dp1</a> (<font color="Maroon" title="c11">m</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c16">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c26">n</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c31">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>) = 
<font color="Maroon" title="c11">m</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c16">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c26">n</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c31">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E24:22_4_5_1_1"><span class="lab"><font color="Green" title="E82">A356</font></span></a>, <a class="ref" href="dualsp05.html#D2" target="_self" onmouseover="rs('dualsp05/D2')" onmouseout="rh()">defDp1</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c9">a</font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c16">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c26">n</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c31">i</font></span>)</span></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> 1</span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p1">(<span class="default"><span class="p2"><a href="rcomp_1.html#K4" title="RCOMP_1:func.4">].</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c16">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c26">n</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c31">i</font></span>)</span>,<font color="Maroon" title="c10">b</font></span><a href="rcomp_1.html#K4" title="RCOMP_1:func.4">.]</a></span> <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:22"><span class="lab"><font color="Green" title="E29">A8</font></span></a>, <a class="txt" href="#E3:22_4_5_1_1_2_1_2"><span class="lab"><font color="Green" title="E87">A377</font></span></a>, <a class="txt" href="#E1:22_4_5_1_1_2_1_2_1_1"><span class="lab"><font color="Green" title="E90">A384</font></span></a>, <a class="txt" href="#E6:22_4_5_1_1_2_1_2"><span class="lab"><font color="Green" title="E89">A382</font></span></a></span>
;<br/><span class="lab"><font color="Green" title="E92">A387</font></span>:  <a href="dualsp05.html#K2" title="DUALSP05:func.2">Dp1</a> (<font color="Maroon" title="c11">m</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c16">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c26">n</font></span>)</span>,<font color="Maroon" title="c31">i</font>) = 
<font color="Maroon" title="c11">m</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c16">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c26">n</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c31">i</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E21:22_4_5_1_1"><span class="lab"><font color="Green" title="E81">A355</font></span></a>, <a class="txt" href="#E1:22_4_5_1_1_2_1_2"><span class="lab"><font color="Green" title="E85">A376</font></span></a>, <a class="ref" href="dualsp05.html#D2" target="_self" onmouseover="rs('dualsp05/D2')" onmouseout="rh()">defDp1</a></span>
<br/>.= 
<span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c9">a</font>,<font color="Maroon" title="c10">b</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:22"><span class="lab"><font color="Green" title="E29">A8</font></span></a>, <a class="txt" href="#E2:22_4_5_1_1_2_1_2"><span class="lab"><font color="Green" title="E86">A378</font></span></a>, <a class="txt" href="#E1:22_4_5_1_1_2_1_2_1_1"><span class="lab"><font color="Green" title="E90">A384</font></span></a></span>
;<br/><a NAME="E5:22_4_5_1_1_2_1_2_1_1"/><span class="lab"><font color="Green" title="E93">A389</font></span>: 
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c9">a</font>,<font color="Maroon" title="c10">b</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">A</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:22"><span class="lab"><font color="Green" title="E28">A7</font></span></a>, <a class="ref" href="funcop_1.html#T13" onmouseover="rs('funcop_1/T13')" onmouseout="rh()">FUNCOP_1:13</a></span>;<br/><a NAME="E6:22_4_5_1_1_2_1_2_1_1"/><span class="lab"><font color="Green" title="E94">A390</font></span>: 
( <font color="Maroon" title="c9">a</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c26">n</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c31">i</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c16">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c26">n</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c31">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c10">b</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:22_4_5_1_1_2_1_2"><span class="lab"><font color="Green" title="E87">A377</font></span></a>, <a class="ref" href="xxreal_1.html#T1" onmouseover="rs('xxreal_1/T1')" onmouseout="rh()">XXREAL_1:1</a></span>;<br/><span class="lab"><font color="Green" title="E95">A392</font></span>:  <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c9">a</font>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c16">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c26">n</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c31">i</font></span>)</span></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> 1</span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p2">(<span class="default"><span class="p3"><a href="rcomp_1.html#K4" title="RCOMP_1:func.4">].</a><span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c16">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c26">n</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c31">i</font></span>)</span>,<font color="Maroon" title="c10">b</font></span><a href="rcomp_1.html#K4" title="RCOMP_1:func.4">.]</a></span> <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p2">(<span class="default"><span class="p3"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c9">a</font>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c16">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c26">n</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c31">i</font></span>)</span></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> 1</span>)</span></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p2">(<span class="default"><span class="p3"><a href="rcomp_1.html#K4" title="RCOMP_1:func.4">].</a><span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c16">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c26">n</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c31">i</font></span>)</span>,<font color="Maroon" title="c10">b</font></span><a href="rcomp_1.html#K4" title="RCOMP_1:func.4">.]</a></span> <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_4.html#D1" onmouseover="rs('funct_4/D1')" onmouseout="rh()">FUNCT_4:def 1</a></span>
<br/>.= 
<font color="Maroon" title="c1">A</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:22"><span class="lab"><font color="Green" title="E28">A7</font></span></a>, <a class="txt" href="#E1:22_4_5_1_1_2"><span class="lab"><font color="Green" title="E84">B20</font></span></a>, <a class="txt" href="#E6:22_4_5_1_1_2_1_2_1_1"><span class="lab"><font color="Green" title="E94">A390</font></span></a>, <a class="ref" href="xxreal_1.html#T167" onmouseover="rs('xxreal_1/T167')" onmouseout="rh()">XXREAL_1:167</a></span>
;<br/><a NAME="E8:22_4_5_1_1_2_1_2_1_1"/>
 <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <span class="p1">(<span class="default"><span class="p2"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c9">a</font>,<font color="Maroon" title="c10">b</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> 
 ;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c43">f0</font> = <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c9">a</font>,<font color="Maroon" title="c10">b</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> as   <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> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:22_4_5_1_1_2_1_2_1_1"><span class="lab"><font color="Green" title="E93">A389</font></span></a>, <a class="ref" href="funct_2.html#T2" onmouseover="rs('funct_2/T2')" onmouseout="rh()">FUNCT_2:2</a></span>;<br/><a NAME="E10:22_4_5_1_1_2_1_2_1_1"/>
 <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c9">a</font>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c16">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c26">n</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c31">i</font></span>)</span></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> 1</span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p2">(<span class="default"><span class="p3"><a href="rcomp_1.html#K4" title="RCOMP_1:func.4">].</a><span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c16">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c26">n</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c31">i</font></span>)</span>,<font color="Maroon" title="c10">b</font></span><a href="rcomp_1.html#K4" title="RCOMP_1:func.4">.]</a></span> <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span>)</span></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> 
 ;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c44">f1</font> = <span class="p1">(<span class="default"><span class="p2"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c9">a</font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c16">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c26">n</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c31">i</font></span>)</span></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> 1</span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p1">(<span class="default"><span class="p2"><a href="rcomp_1.html#K4" title="RCOMP_1:func.4">].</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c16">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c26">n</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c31">i</font></span>)</span>,<font color="Maroon" title="c10">b</font></span><a href="rcomp_1.html#K4" title="RCOMP_1:func.4">.]</a></span> <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span>)</span> as   <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> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:22_4_5_1_1_2_1_2_1_1"><span class="lab"><font color="Green" title="E95">A392</font></span></a>, <a class="ref" href="funct_2.html#T2" onmouseover="rs('funct_2/T2')" onmouseout="rh()">FUNCT_2:2</a></span>;<br/><a NAME="E12:22_4_5_1_1_2_1_2_1_1"/><span class="lab"><font color="Green" title="E96">A394</font></span>: 
<font color="Maroon" title="c34">qi</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c30">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c32">r</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c44">f1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c30">t</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c43">f0</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c30">t</font></span>)</span></span>)</span>
 <div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="22_4_5_1_1_2_1_2_1_1_4"><span class="kw">proof </span></a><div class="add">

<span class="kw">reconsider </span><font color="Maroon" title="c45">H</font> = <span class="p1">(<span class="default"><a href="dualsp05.html#K2" title="DUALSP05:func.2">Dp1</a> (<font color="Maroon" title="c11">m</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c16">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c26">n</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c31">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>)</span>)</span> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <span class="p1">(<span class="default"><a href="dualsp05.html#K2" title="DUALSP05:func.2">Dp1</a> (<font color="Maroon" title="c11">m</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c16">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c26">n</font></span>)</span>,<font color="Maroon" title="c31">i</font>)</span>)</span> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="c0sp1.html#K11" title="C0SP1:func.11">R_Normed_Algebra_of_BoundedFunctions</a> <font color="Maroon" title="c1">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="c46">h</font> = <font color="Maroon" title="c45">H</font> as   <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> <span class="kw">by</span> <span class="lab"><a class="ref" href="dualsp05.html#T11" target="_self" onmouseover="rs('dualsp05/T11')" onmouseout="rh()">LM88</a></span>;<br/>
<a NAME="E3:22_4_5_1_1_2_1_2_1_1_4"/>
<font color="Maroon" title="c34">qi</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c32">r</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c45">H</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="dualsp05.html#T5" target="_self" onmouseover="rs('dualsp05/T5')" onmouseout="rh()">Lm1</a>, <a class="txt" href="#E17:22_4_5_1_1"><span class="lab"><font color="Green" title="E78">A351</font></span></a>, <a class="txt" href="#E7:22_4_5_1_1"><span class="lab"><font color="Green" title="E70">A338</font></span></a></span>;<br/>
<span class="kw">then </span><font color="Maroon" title="c34">qi</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c30">t</font> = 
<font color="Maroon" title="c32">r</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c46">h</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c30">t</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="c0sp1.html#T30" onmouseover="rs('c0sp1/T30')" onmouseout="rh()">C0SP1:30</a></span>
<br/>.= 
<font color="Maroon" title="c32">r</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c44">f1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c30">t</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c43">f0</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c30">t</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:22"><span class="lab"><font color="Green" title="E23">A2</font></span></a>, <a class="txt" href="#E3:22_4_5_1_1_2_1_2_1_1"><span class="lab"><font color="Green" title="E91">A386</font></span></a>, <a class="txt" href="#E4:22_4_5_1_1_2_1_2_1_1"><span class="lab"><font color="Green" title="E92">A387</font></span></a>, <a class="ref" href="c0sp1.html#T34" onmouseover="rs('c0sp1/T34')" onmouseout="rh()">C0SP1:34</a></span>
;<br/>
<span class="kw">hence </span><a NAME="E5:22_4_5_1_1_2_1_2_1_1_4"/>
<font color="Maroon" title="c34">qi</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c30">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c32">r</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c44">f1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c30">t</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c43">f0</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c30">t</font></span>)</span></span>)</span>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><a NAME="E13:22_4_5_1_1_2_1_2_1_1"/><span class="lab"><font color="Green" title="E97">A395</font></span>: 
<font color="Maroon" title="c30">t</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c16">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c26">n</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c31">i</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c16">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c26">n</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c31">i</font></span>)</span></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:22_4_5_1_1_2_1_2"><span class="lab"><font color="Green" title="E88">A379</font></span></a>, <a class="ref" href="integra1.html#T4" onmouseover="rs('integra1/T4')" onmouseout="rh()">INTEGRA1:4</a>, <a class="txt" href="#E2:22_4_5_1_1"><span class="lab"><font color="Green" title="E66">A333</font></span></a></span>;<br/><a NAME="E14:22_4_5_1_1_2_1_2_1_1"/>
( <font color="Maroon" title="c9">a</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c26">n</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c31">i</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c16">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c26">n</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c31">i</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c10">b</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:22_4_5_1_1_2_1_2"><span class="lab"><font color="Green" title="E86">A378</font></span></a>, <a class="ref" href="xxreal_1.html#T1" onmouseover="rs('xxreal_1/T1')" onmouseout="rh()">XXREAL_1:1</a></span>;<br/><a NAME="E15:22_4_5_1_1_2_1_2_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E98">B396</font></span>: 
<span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c16">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c26">n</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c31">i</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c16">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c26">n</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c31">i</font></span>)</span></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c9">a</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c16">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c26">n</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c31">i</font></span>)</span></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_1.html#T34" onmouseover="rs('xxreal_1/T34')" onmouseout="rh()">XXREAL_1:34</a></span>;<br/><span class="kw">then </span><span class="lab"><font color="Green" title="E99">A398</font></span>: <font color="Maroon" title="c44">f1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c30">t</font> = 
<span class="p1">(<span class="default"><span class="p2"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c9">a</font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c16">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c26">n</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c31">i</font></span>)</span></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> 1</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c30">t</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:22_4_5_1_1_2_1_2_1_1"><span class="lab"><font color="Green" title="E97">A395</font></span></a>, <a class="ref" href="funct_4.html#T16" onmouseover="rs('funct_4/T16')" onmouseout="rh()">FUNCT_4:16</a>, <a class="txt" href="#E1:22_4_5_1_1_2"><span class="lab"><font color="Green" title="E84">B20</font></span></a>, <a class="ref" href="xxreal_1.html#T90" onmouseover="rs('xxreal_1/T90')" onmouseout="rh()">XXREAL_1:90</a></span>
<br/>.= 
1
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:22_4_5_1_1_2_1_2_1_1"><span class="lab"><font color="Green" title="E98">B396</font></span></a>, <a class="txt" href="#E13:22_4_5_1_1_2_1_2_1_1"><span class="lab"><font color="Green" title="E97">A395</font></span></a>, <a class="ref" href="funcop_1.html#T7" onmouseover="rs('funcop_1/T7')" onmouseout="rh()">FUNCOP_1:7</a></span>
;<br/><a NAME="E17:22_4_5_1_1_2_1_2_1_1"/>
<font color="Maroon" title="c43">f0</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c30">t</font> <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="txt" href="#E11:22"><span class="lab"><font color="Green" title="E28">A7</font></span></a>, <a class="ref" href="funcop_1.html#T7" onmouseover="rs('funcop_1/T7')" onmouseout="rh()">FUNCOP_1:7</a></span>;<br/><span class="kw">hence </span><a NAME="E18:22_4_5_1_1_2_1_2_1_1"/>
<font color="Maroon" title="c33">z</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c31">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c32">r</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:22_4_5_1_1"><span class="lab"><font color="Green" title="E78">A351</font></span></a>, <a class="txt" href="#E12:22_4_5_1_1_2_1_2_1_1"><span class="lab"><font color="Green" title="E96">A394</font></span></a>, <a class="txt" href="#E16:22_4_5_1_1_2_1_2_1_1"><span class="lab"><font color="Green" title="E99">A398</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><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="22_4_5_1_1_2_1_2_1_2"><span class="kw">suppose </span></a><a NAME="E1:22_4_5_1_1_2_1_2_1_2"/><span class="lab"><font color="Green" title="E90">A399</font></span>: 
<font color="Maroon" title="c9">a</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c26">n</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c31">i</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</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="c33">z</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c31">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c32">r</font></span><br/><div class="add"><a NAME="E2:22_4_5_1_1_2_1_2_1_2"/>
1 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c31">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T8" onmouseover="rs('xreal_1/T8')" onmouseout="rh()">XREAL_1:8</a>, <a class="txt" href="#E19:22_4_5_1_1"><span class="lab"><font color="Green" title="E80">A354</font></span></a></span>;<br/><span class="kw">then </span><span class="lab"><font color="Green" title="E91">A401</font></span>:  <a href="dualsp05.html#K2" title="DUALSP05:func.2">Dp1</a> (<font color="Maroon" title="c11">m</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c16">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c26">n</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c31">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>) = 
<font color="Maroon" title="c11">m</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c16">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c26">n</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c31">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E24:22_4_5_1_1"><span class="lab"><font color="Green" title="E82">A356</font></span></a>, <a class="ref" href="dualsp05.html#D2" target="_self" onmouseover="rs('dualsp05/D2')" onmouseout="rh()">defDp1</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c9">a</font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c16">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c26">n</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c31">i</font></span>)</span></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> 1</span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p1">(<span class="default"><span class="p2"><a href="rcomp_1.html#K4" title="RCOMP_1:func.4">].</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c16">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c26">n</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c31">i</font></span>)</span>,<font color="Maroon" title="c10">b</font></span><a href="rcomp_1.html#K4" title="RCOMP_1:func.4">.]</a></span> <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:22"><span class="lab"><font color="Green" title="E29">A8</font></span></a>, <a class="txt" href="#E3:22_4_5_1_1_2_1_2"><span class="lab"><font color="Green" title="E87">A377</font></span></a>, <a class="txt" href="#E1:22_4_5_1_1_2_1_2_1_2"><span class="lab"><font color="Green" title="E90">A399</font></span></a>, <a class="txt" href="#E6:22_4_5_1_1_2_1_2"><span class="lab"><font color="Green" title="E89">A382</font></span></a></span>
;<br/><span class="lab"><font color="Green" title="E92">A402</font></span>:  <a href="dualsp05.html#K2" title="DUALSP05:func.2">Dp1</a> (<font color="Maroon" title="c11">m</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c16">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c26">n</font></span>)</span>,<font color="Maroon" title="c31">i</font>) = 
<font color="Maroon" title="c11">m</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c16">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c26">n</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c31">i</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E21:22_4_5_1_1"><span class="lab"><font color="Green" title="E81">A355</font></span></a>, <a class="txt" href="#E1:22_4_5_1_1_2_1_2"><span class="lab"><font color="Green" title="E85">A376</font></span></a>, <a class="ref" href="dualsp05.html#D2" target="_self" onmouseover="rs('dualsp05/D2')" onmouseout="rh()">defDp1</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c9">a</font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c16">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c26">n</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c31">i</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span></span>)</span></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> 1</span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p1">(<span class="default"><span class="p2"><a href="rcomp_1.html#K4" title="RCOMP_1:func.4">].</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c16">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c26">n</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c31">i</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c10">b</font></span><a href="rcomp_1.html#K4" title="RCOMP_1:func.4">.]</a></span> <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:22"><span class="lab"><font color="Green" title="E29">A8</font></span></a>, <a class="txt" href="#E2:22_4_5_1_1_2_1_2"><span class="lab"><font color="Green" title="E86">A378</font></span></a>, <a class="txt" href="#E1:22_4_5_1_1_2_1_2_1_2"><span class="lab"><font color="Green" title="E90">A399</font></span></a></span>
;<br/><a NAME="E5:22_4_5_1_1_2_1_2_1_2"/><span class="lab"><font color="Green" title="E93">A404</font></span>: 
( <font color="Maroon" title="c9">a</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c26">n</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c31">i</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c16">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c26">n</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c31">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c10">b</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:22_4_5_1_1_2_1_2"><span class="lab"><font color="Green" title="E87">A377</font></span></a>, <a class="ref" href="xxreal_1.html#T1" onmouseover="rs('xxreal_1/T1')" onmouseout="rh()">XXREAL_1:1</a></span>;<br/><a NAME="E6:22_4_5_1_1_2_1_2_1_2"/><span class="lab"><font color="Green" title="E94">A405</font></span>: 
( <font color="Maroon" title="c9">a</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c26">n</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c31">i</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c16">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c26">n</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c31">i</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c10">b</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:22_4_5_1_1_2_1_2"><span class="lab"><font color="Green" title="E86">A378</font></span></a>, <a class="ref" href="xxreal_1.html#T1" onmouseover="rs('xxreal_1/T1')" onmouseout="rh()">XXREAL_1:1</a></span>;<br/><span class="lab"><font color="Green" title="E95">A406</font></span>:  <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c9">a</font>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c16">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c26">n</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c31">i</font></span>)</span></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> 1</span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p2">(<span class="default"><span class="p3"><a href="rcomp_1.html#K4" title="RCOMP_1:func.4">].</a><span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c16">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c26">n</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c31">i</font></span>)</span>,<font color="Maroon" title="c10">b</font></span><a href="rcomp_1.html#K4" title="RCOMP_1:func.4">.]</a></span> <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p2">(<span class="default"><span class="p3"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c9">a</font>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c16">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c26">n</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c31">i</font></span>)</span></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> 1</span>)</span></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p2">(<span class="default"><span class="p3"><a href="rcomp_1.html#K4" title="RCOMP_1:func.4">].</a><span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c16">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c26">n</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c31">i</font></span>)</span>,<font color="Maroon" title="c10">b</font></span><a href="rcomp_1.html#K4" title="RCOMP_1:func.4">.]</a></span> <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_4.html#D1" onmouseover="rs('funct_4/D1')" onmouseout="rh()">FUNCT_4:def 1</a></span>
<br/>.= 
<font color="Maroon" title="c1">A</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:22"><span class="lab"><font color="Green" title="E28">A7</font></span></a>, <a class="txt" href="#E1:22_4_5_1_1_2"><span class="lab"><font color="Green" title="E84">B20</font></span></a>, <a class="txt" href="#E5:22_4_5_1_1_2_1_2_1_2"><span class="lab"><font color="Green" title="E93">A404</font></span></a>, <a class="ref" href="xxreal_1.html#T167" onmouseover="rs('xxreal_1/T167')" onmouseout="rh()">XXREAL_1:167</a></span>
;<br/><span class="lab"><font color="Green" title="E96">A407</font></span>:  <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c9">a</font>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c16">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c26">n</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p5">(<span class="default"><font color="Maroon" title="c31">i</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span></span>)</span></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> 1</span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p2">(<span class="default"><span class="p3"><a href="rcomp_1.html#K4" title="RCOMP_1:func.4">].</a><span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c16">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c26">n</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p5">(<span class="default"><font color="Maroon" title="c31">i</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c10">b</font></span><a href="rcomp_1.html#K4" title="RCOMP_1:func.4">.]</a></span> <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p2">(<span class="default"><span class="p3"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c9">a</font>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c16">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c26">n</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p5">(<span class="default"><font color="Maroon" title="c31">i</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span></span>)</span></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> 1</span>)</span></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p2">(<span class="default"><span class="p3"><a href="rcomp_1.html#K4" title="RCOMP_1:func.4">].</a><span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c16">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c26">n</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p5">(<span class="default"><font color="Maroon" title="c31">i</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c10">b</font></span><a href="rcomp_1.html#K4" title="RCOMP_1:func.4">.]</a></span> <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_4.html#D1" onmouseover="rs('funct_4/D1')" onmouseout="rh()">FUNCT_4:def 1</a></span>
<br/>.= 
<font color="Maroon" title="c1">A</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:22"><span class="lab"><font color="Green" title="E28">A7</font></span></a>, <a class="txt" href="#E1:22_4_5_1_1_2"><span class="lab"><font color="Green" title="E84">B20</font></span></a>, <a class="txt" href="#E6:22_4_5_1_1_2_1_2_1_2"><span class="lab"><font color="Green" title="E94">A405</font></span></a>, <a class="ref" href="xxreal_1.html#T167" onmouseover="rs('xxreal_1/T167')" onmouseout="rh()">XXREAL_1:167</a></span>
;<br/><a NAME="E9:22_4_5_1_1_2_1_2_1_2"/>
 <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c9">a</font>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c16">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c26">n</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c31">i</font></span>)</span></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> 1</span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p2">(<span class="default"><span class="p3"><a href="rcomp_1.html#K4" title="RCOMP_1:func.4">].</a><span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c16">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c26">n</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c31">i</font></span>)</span>,<font color="Maroon" title="c10">b</font></span><a href="rcomp_1.html#K4" title="RCOMP_1:func.4">.]</a></span> <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span>)</span></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> 
 ;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c43">f1</font> = <span class="p1">(<span class="default"><span class="p2"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c9">a</font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c16">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c26">n</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c31">i</font></span>)</span></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> 1</span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p1">(<span class="default"><span class="p2"><a href="rcomp_1.html#K4" title="RCOMP_1:func.4">].</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c16">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c26">n</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c31">i</font></span>)</span>,<font color="Maroon" title="c10">b</font></span><a href="rcomp_1.html#K4" title="RCOMP_1:func.4">.]</a></span> <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span>)</span> as   <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> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:22_4_5_1_1_2_1_2_1_2"><span class="lab"><font color="Green" title="E95">A406</font></span></a>, <a class="ref" href="funct_2.html#T2" onmouseover="rs('funct_2/T2')" onmouseout="rh()">FUNCT_2:2</a></span>;<br/><a NAME="E11:22_4_5_1_1_2_1_2_1_2"/>
 <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c9">a</font>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c16">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c26">n</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p5">(<span class="default"><font color="Maroon" title="c31">i</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span></span>)</span></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> 1</span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p2">(<span class="default"><span class="p3"><a href="rcomp_1.html#K4" title="RCOMP_1:func.4">].</a><span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c16">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c26">n</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p5">(<span class="default"><font color="Maroon" title="c31">i</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c10">b</font></span><a href="rcomp_1.html#K4" title="RCOMP_1:func.4">.]</a></span> <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span>)</span></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> 
 ;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c44">f2</font> = <span class="p1">(<span class="default"><span class="p2"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c9">a</font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c16">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c26">n</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c31">i</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span></span>)</span></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> 1</span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p1">(<span class="default"><span class="p2"><a href="rcomp_1.html#K4" title="RCOMP_1:func.4">].</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c16">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c26">n</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c31">i</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c10">b</font></span><a href="rcomp_1.html#K4" title="RCOMP_1:func.4">.]</a></span> <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span>)</span> as   <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> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:22_4_5_1_1_2_1_2_1_2"><span class="lab"><font color="Green" title="E96">A407</font></span></a>, <a class="ref" href="funct_2.html#T2" onmouseover="rs('funct_2/T2')" onmouseout="rh()">FUNCT_2:2</a></span>;<br/><a NAME="E13:22_4_5_1_1_2_1_2_1_2"/><span class="lab"><font color="Green" title="E97">A408</font></span>: 
( <font color="Maroon" title="c9">a</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c30">t</font> &amp; <font color="Maroon" title="c30">t</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c10">b</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_1.html#T1" onmouseover="rs('xxreal_1/T1')" onmouseout="rh()">XXREAL_1:1</a>, <a class="txt" href="#E11:22"><span class="lab"><font color="Green" title="E28">A7</font></span></a></span>;<br/><a NAME="E14:22_4_5_1_1_2_1_2_1_2"/><span class="lab"><font color="Green" title="E98">A409</font></span>: 
<font color="Maroon" title="c34">qi</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c30">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c32">r</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c43">f1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c30">t</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c44">f2</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c30">t</font></span>)</span></span>)</span>
 <div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="22_4_5_1_1_2_1_2_1_2_5"><span class="kw">proof </span></a><div class="add">

<span class="kw">reconsider </span><font color="Maroon" title="c45">H</font> = <span class="p1">(<span class="default"><a href="dualsp05.html#K2" title="DUALSP05:func.2">Dp1</a> (<font color="Maroon" title="c11">m</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c16">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c26">n</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c31">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>)</span>)</span> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <span class="p1">(<span class="default"><a href="dualsp05.html#K2" title="DUALSP05:func.2">Dp1</a> (<font color="Maroon" title="c11">m</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c16">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c26">n</font></span>)</span>,<font color="Maroon" title="c31">i</font>)</span>)</span> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="c0sp1.html#K11" title="C0SP1:func.11">R_Normed_Algebra_of_BoundedFunctions</a> <font color="Maroon" title="c1">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="c46">h</font> = <font color="Maroon" title="c45">H</font> as   <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> <span class="kw">by</span> <span class="lab"><a class="ref" href="dualsp05.html#T11" target="_self" onmouseover="rs('dualsp05/T11')" onmouseout="rh()">LM88</a></span>;<br/>
<a NAME="E3:22_4_5_1_1_2_1_2_1_2_5"/>
<font color="Maroon" title="c34">qi</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c32">r</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c45">H</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="dualsp05.html#T5" target="_self" onmouseover="rs('dualsp05/T5')" onmouseout="rh()">Lm1</a>, <a class="txt" href="#E17:22_4_5_1_1"><span class="lab"><font color="Green" title="E78">A351</font></span></a>, <a class="txt" href="#E7:22_4_5_1_1"><span class="lab"><font color="Green" title="E70">A338</font></span></a></span>;<br/>
<span class="kw">then </span><font color="Maroon" title="c34">qi</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c30">t</font> = 
<font color="Maroon" title="c32">r</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c46">h</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c30">t</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="c0sp1.html#T30" onmouseover="rs('c0sp1/T30')" onmouseout="rh()">C0SP1:30</a></span>
<br/>.= 
<font color="Maroon" title="c32">r</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c43">f1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c30">t</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c44">f2</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c30">t</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:22"><span class="lab"><font color="Green" title="E23">A2</font></span></a>, <a class="txt" href="#E3:22_4_5_1_1_2_1_2_1_2"><span class="lab"><font color="Green" title="E91">A401</font></span></a>, <a class="txt" href="#E4:22_4_5_1_1_2_1_2_1_2"><span class="lab"><font color="Green" title="E92">A402</font></span></a>, <a class="ref" href="c0sp1.html#T34" onmouseover="rs('c0sp1/T34')" onmouseout="rh()">C0SP1:34</a></span>
;<br/>
<span class="kw">hence </span><a NAME="E5:22_4_5_1_1_2_1_2_1_2_5"/>
<font color="Maroon" title="c34">qi</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c30">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c32">r</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c43">f1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c30">t</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c44">f2</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c30">t</font></span>)</span></span>)</span>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><a NAME="E15:22_4_5_1_1_2_1_2_1_2"/><span class="lab"><font color="Green" title="E99">A410</font></span>: 
<font color="Maroon" title="c30">t</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c16">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c26">n</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c31">i</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c16">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c26">n</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c31">i</font></span>)</span></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:22_4_5_1_1_2_1_2"><span class="lab"><font color="Green" title="E88">A379</font></span></a>, <a class="ref" href="integra1.html#T4" onmouseover="rs('integra1/T4')" onmouseout="rh()">INTEGRA1:4</a>, <a class="txt" href="#E2:22_4_5_1_1"><span class="lab"><font color="Green" title="E66">A333</font></span></a></span>;<br/><a NAME="E16:22_4_5_1_1_2_1_2_1_2"/><span class="lab"><font color="Green" title="E100">B411</font></span>: 
<span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c16">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c26">n</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c31">i</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c16">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c26">n</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c31">i</font></span>)</span></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c9">a</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c16">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c26">n</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c31">i</font></span>)</span></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:22_4_5_1_1_2_1_2_1_2"><span class="lab"><font color="Green" title="E94">A405</font></span></a>, <a class="ref" href="xxreal_1.html#T34" onmouseover="rs('xxreal_1/T34')" onmouseout="rh()">XXREAL_1:34</a></span>;<br/><span class="kw">then </span><span class="lab"><font color="Green" title="E101">A413</font></span>: <font color="Maroon" title="c43">f1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c30">t</font> = 
<span class="p1">(<span class="default"><span class="p2"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c9">a</font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c16">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c26">n</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c31">i</font></span>)</span></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> 1</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c30">t</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:22_4_5_1_1_2_1_2_1_2"><span class="lab"><font color="Green" title="E99">A410</font></span></a>, <a class="ref" href="funct_4.html#T16" onmouseover="rs('funct_4/T16')" onmouseout="rh()">FUNCT_4:16</a>, <a class="txt" href="#E1:22_4_5_1_1_2"><span class="lab"><font color="Green" title="E84">B20</font></span></a>, <a class="ref" href="xxreal_1.html#T90" onmouseover="rs('xxreal_1/T90')" onmouseout="rh()">XXREAL_1:90</a></span>
<br/>.= 
1
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:22_4_5_1_1_2_1_2_1_2"><span class="lab"><font color="Green" title="E100">B411</font></span></a>, <a class="txt" href="#E15:22_4_5_1_1_2_1_2_1_2"><span class="lab"><font color="Green" title="E99">A410</font></span></a>, <a class="ref" href="funcop_1.html#T7" onmouseover="rs('funcop_1/T7')" onmouseout="rh()">FUNCOP_1:7</a></span>
;<br/><a NAME="E18:22_4_5_1_1_2_1_2_1_2"/>
( <span class="p1">(<span class="default"><font color="Maroon" title="c16">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c26">n</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c31">i</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c30">t</font> &amp; <font color="Maroon" title="c30">t</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c26">n</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c31">i</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:22_4_5_1_1"><span class="lab"><font color="Green" title="E65">A332</font></span></a>, <a class="ref" href="integra1.html#D4" onmouseover="rs('integra1/D4')" onmouseout="rh()">INTEGRA1:def 4</a>, <a class="txt" href="#E3:22_4_5_1_1"><span class="lab"><font color="Green" title="E67">A334</font></span></a>, <a class="txt" href="#E1:22_4_5_1_1_2_1_2"><span class="lab"><font color="Green" title="E85">A376</font></span></a></span>;<br/><a NAME="E19:22_4_5_1_1_2_1_2_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E102">A414</font></span>: 
<font color="Maroon" title="c30">t</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"><a href="rcomp_1.html#K4" title="RCOMP_1:func.4">].</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c16">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c26">n</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c31">i</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c10">b</font></span><a href="rcomp_1.html#K4" title="RCOMP_1:func.4">.]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:22_4_5_1_1_2_1_2_1_2"><span class="lab"><font color="Green" title="E97">A408</font></span></a></span>;<br/><span class="kw">then </span><font color="Maroon" title="c44">f2</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c30">t</font> = 
<span class="p1">(<span class="default"><span class="p2"><a href="rcomp_1.html#K4" title="RCOMP_1:func.4">].</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c16">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c26">n</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c31">i</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c10">b</font></span><a href="rcomp_1.html#K4" title="RCOMP_1:func.4">.]</a></span> <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c30">t</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:22_4_5_1_1_2"><span class="lab"><font color="Green" title="E84">B20</font></span></a>, <a class="ref" href="funct_4.html#T13" onmouseover="rs('funct_4/T13')" onmouseout="rh()">FUNCT_4:13</a></span>
<br/>.= 
 <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:22_4_5_1_1_2_1_2_1_2"><span class="lab"><font color="Green" title="E102">A414</font></span></a>, <a class="ref" href="funcop_1.html#T7" onmouseover="rs('funcop_1/T7')" onmouseout="rh()">FUNCOP_1:7</a></span>
;<br/><span class="kw">hence </span><a NAME="E21:22_4_5_1_1_2_1_2_1_2"/>
<font color="Maroon" title="c33">z</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c31">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c32">r</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:22_4_5_1_1"><span class="lab"><font color="Green" title="E78">A351</font></span></a>, <a class="txt" href="#E14:22_4_5_1_1_2_1_2_1_2"><span class="lab"><font color="Green" title="E98">A409</font></span></a>, <a class="txt" href="#E17:22_4_5_1_1_2_1_2_1_2"><span class="lab"><font color="Green" title="E101">A413</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></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

</div><span class="kw">end;</span></div>
<a NAME="E26:22_4_5_1_1"/>
 for <font color="Olive" title="b1">k</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Olive" title="b1">k</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> <font color="Maroon" title="c33">z</font> &amp; <font color="Olive" title="b1">k</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c31">i</font> holds <br/><font color="Maroon" title="c33">z</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> 
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="22_4_5_1_1_3"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c36">k</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="c36">k</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> <font color="Maroon" title="c33">z</font> &amp; <font color="Maroon" title="c36">k</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c31">i</font> implies <font color="Maroon" title="c33">z</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c36">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:22_4_5_1_1_3"/><span class="lab"><font color="Green" title="E84">A415</font></span>: 
<font color="Maroon" title="c36">k</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> <font color="Maroon" title="c33">z</font>
 <span class="kw">and </span><br/><a NAME="E2:22_4_5_1_1_3"/><span class="lab"><font color="Green" title="E85">A416</font></span>: 
<font color="Maroon" title="c36">k</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c31">i</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="c33">z</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c36">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> </span><br/>

<span class="kw">consider </span><font color="Maroon" title="c37">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="E4:22_4_5_1_1_3"/>
<font color="Maroon" title="c27">p2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c36">k</font> <a href="tarski.html#R2" title="TARSKI: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="c15">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="c16">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c26">n</font></span>)</span>,<font color="Maroon" title="c36">k</font>)</span>)</span></span>)</span>
 <span class="kw">and </span><br/><a NAME="E5:22_4_5_1_1_3"/>
<font color="Maroon" title="c37">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">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="c16">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c26">n</font></span>)</span>,<font color="Maroon" title="c36">k</font>)</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c27">p2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c36">k</font></span>)</span>
 <span class="kw">and </span><br/><a NAME="E6:22_4_5_1_1_3"/><span class="lab"><font color="Green" title="E86">A420</font></span>: 
<font color="Maroon" title="c28">q2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c36">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c37">r</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="dualsp05.html#K2" title="DUALSP05:func.2">Dp1</a> (<font color="Maroon" title="c11">m</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c16">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c26">n</font></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon" title="c36">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>)</span>)</span> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <span class="p2">(<span class="default"><a href="dualsp05.html#K2" title="DUALSP05:func.2">Dp1</a> (<font color="Maroon" title="c11">m</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c16">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c26">n</font></span>)</span>,<font color="Maroon" title="c36">k</font>)</span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:22_4_5_1"><span class="lab"><font color="Green" title="E60">A327</font></span></a>, <a class="txt" href="#E5:22_4_5_1"><span class="lab"><font color="Green" title="E61">A328</font></span></a>, <a class="txt" href="#E14:22_4_5_1_1"><span class="lab"><font color="Green" title="E76">A349</font></span></a>, <a class="txt" href="#E1:22_4_5_1_1_3"><span class="lab"><font color="Green" title="E84">A415</font></span></a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c38">qk</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><span class="kw"> such that </span><br/><a NAME="E8:22_4_5_1_1_3"/><span class="lab"><font color="Green" title="E87">A423</font></span>: 
( <font color="Maroon" title="c38">qk</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c28">q2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c36">k</font> &amp; <font color="Maroon" title="c33">z</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c36">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c38">qk</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c30">t</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:22_4_5_1_1"><span class="lab"><font color="Green" title="E72">A344</font></span></a>, <a class="txt" href="#E1:22_4_5_1_1_3"><span class="lab"><font color="Green" title="E84">A415</font></span></a></span>;<br/>
<a NAME="E9:22_4_5_1_1_3"/><span class="lab"><font color="Green" title="E88">A425</font></span>: 
<font color="Maroon" title="c36">k</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="c16">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c26">n</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:22_4_5_1_1"><span class="lab"><font color="Green" title="E72">A344</font></span></a>, <a class="txt" href="#E1:22_4_5_1_1_3"><span class="lab"><font color="Green" title="E84">A415</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="E10:22_4_5_1_1_3"/><span class="lab"><font color="Green" title="E89">A426</font></span>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c36">k</font> &amp; <font color="Maroon" title="c36">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c26">n</font></span>)</span> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:22_4_5_1_1"><span class="lab"><font color="Green" title="E72">A344</font></span></a>, <a class="txt" href="#E1:22_4_5_1_1_3"><span class="lab"><font color="Green" title="E84">A415</font></span></a>, <a class="ref" href="finseq_1.html#T1" onmouseover="rs('finseq_1/T1')" onmouseout="rh()">FINSEQ_1:1</a></span>;<br/>
<a NAME="E11:22_4_5_1_1_3"/><span class="kw">then </span>
<font color="Maroon" title="c36">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</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="c16">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c26">n</font></span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T7" onmouseover="rs('xreal_1/T7')" onmouseout="rh()">XREAL_1:7</a></span>;<br/>
<a NAME="E12:22_4_5_1_1_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E90">A427</font></span>: 
<font color="Maroon" title="c36">k</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"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c16">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c26">n</font></span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:22_4_5_1_1_3"><span class="lab"><font color="Green" title="E89">A426</font></span></a></span>;<br/>
<a NAME="E13:22_4_5_1_1_3"/><span class="lab"><font color="Green" title="E91">e</font></span>: 
1 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c36">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T7" onmouseover="rs('xreal_1/T7')" onmouseout="rh()">XREAL_1:7</a></span>;<br/>
<a NAME="E14:22_4_5_1_1_3"/>
<font color="Maroon" title="c36">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</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="c16">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c26">n</font></span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T7" onmouseover="rs('xreal_1/T7')" onmouseout="rh()">XREAL_1:7</a>, <a class="txt" href="#E10:22_4_5_1_1_3"><span class="lab"><font color="Green" title="E89">A426</font></span></a></span>;<br/>
<a NAME="E15:22_4_5_1_1_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E92">A428</font></span>: 
<font color="Maroon" title="c36">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <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"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c16">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c26">n</font></span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:22_4_5_1_1_3"><span class="lab"><font color="Green" title="E91">e</font></span></a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c39">f0</font> = <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c9">a</font>,<font color="Maroon" title="c10">b</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>;<br/>
<span class="kw">set </span><font color="Maroon" title="c40">f1</font> = <span class="p1">(<span class="default"><span class="p2"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c9">a</font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c16">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c26">n</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c36">k</font></span>)</span></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> 1</span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p1">(<span class="default"><span class="p2"><a href="rcomp_1.html#K4" title="RCOMP_1:func.4">].</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c16">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c26">n</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c36">k</font></span>)</span>,<font color="Maroon" title="c10">b</font></span><a href="rcomp_1.html#K4" title="RCOMP_1:func.4">.]</a></span> <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span>)</span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c41">g1</font> = <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c9">a</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c16">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c26">n</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c36">k</font></span>)</span></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> 1;<br/>
<span class="kw">set </span><font color="Maroon" title="c42">g2</font> = <span class="p1"><a href="rcomp_1.html#K4" title="RCOMP_1:func.4">].</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c16">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c26">n</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c36">k</font></span>)</span>,<font color="Maroon" title="c10">b</font></span><a href="rcomp_1.html#K4" title="RCOMP_1:func.4">.]</a></span> <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>;<br/>
<span class="kw">set </span><font color="Maroon" title="c43">f2</font> = <span class="p1">(<span class="default"><span class="p2"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c9">a</font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c16">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c26">n</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c36">k</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span></span>)</span></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> 1</span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p1">(<span class="default"><span class="p2"><a href="rcomp_1.html#K4" title="RCOMP_1:func.4">].</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c16">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c26">n</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c36">k</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c10">b</font></span><a href="rcomp_1.html#K4" title="RCOMP_1:func.4">.]</a></span> <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span>)</span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c44">h1</font> = <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c9">a</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c16">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c26">n</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c36">k</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span></span>)</span></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> 1;<br/>
<span class="kw">set </span><font color="Maroon" title="c45">h2</font> = <span class="p1"><a href="rcomp_1.html#K4" title="RCOMP_1:func.4">].</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c16">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c26">n</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c36">k</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c10">b</font></span><a href="rcomp_1.html#K4" title="RCOMP_1:func.4">.]</a></span> <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>;<br/>
<a NAME="E16:22_4_5_1_1_3"/><span class="lab"><font color="Green" title="E93">B30</font></span>: 
(  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c9">a</font>,<font color="Maroon" title="c10">b</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c9">a</font>,<font color="Maroon" title="c10">b</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> &amp;  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c9">a</font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c16">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c26">n</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c36">k</font></span>)</span></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c9">a</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c16">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c26">n</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c36">k</font></span>)</span></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> &amp;  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2"><a href="rcomp_1.html#K4" title="RCOMP_1:func.4">].</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c16">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c26">n</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c36">k</font></span>)</span>,<font color="Maroon" title="c10">b</font></span><a href="rcomp_1.html#K4" title="RCOMP_1:func.4">.]</a></span> <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="rcomp_1.html#K4" title="RCOMP_1:func.4">].</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c16">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c26">n</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c36">k</font></span>)</span>,<font color="Maroon" title="c10">b</font></span><a href="rcomp_1.html#K4" title="RCOMP_1:func.4">.]</a></span> &amp;  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c9">a</font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c16">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c26">n</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c36">k</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span></span>)</span></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c9">a</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c16">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c26">n</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c36">k</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span></span>)</span></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> &amp;  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2"><a href="rcomp_1.html#K4" title="RCOMP_1:func.4">].</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c16">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c26">n</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c36">k</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c10">b</font></span><a href="rcomp_1.html#K4" title="RCOMP_1:func.4">.]</a></span> <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="rcomp_1.html#K4" title="RCOMP_1:func.4">].</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c16">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c26">n</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c36">k</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c10">b</font></span><a href="rcomp_1.html#K4" title="RCOMP_1:func.4">.]</a></span> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funcop_1.html#T13" onmouseover="rs('funcop_1/T13')" onmouseout="rh()">FUNCOP_1:13</a></span>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="22_4_5_1_1_3_1"><span class="kw">per </span></a><a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:22_4_5_1_1_3_1"/>
( <font color="Maroon" title="c36">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or <font color="Maroon" title="c36">k</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 1 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="22_4_5_1_1_3_1_1"><span class="kw">suppose </span></a><a NAME="E1:22_4_5_1_1_3_1_1"/><span class="lab"><font color="Green" title="E94">A429</font></span>: 
<font color="Maroon" title="c36">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 ; <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="c33">z</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c36">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> </span><br/><div class="add"><a NAME="E2:22_4_5_1_1_3_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E95">A433</font></span>: 
(  <a href="seq_4.html#K3" title="SEQ_4:func.3">lower_bound</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c16">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c26">n</font></span>)</span>,<font color="Maroon" title="c31">i</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c26">n</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c31">i</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span> &amp;  <a href="seq_4.html#K2" title="SEQ_4:func.2">upper_bound</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c16">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c26">n</font></span>)</span>,<font color="Maroon" title="c31">i</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c26">n</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c31">i</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:22_4_5_1_1"><span class="lab"><font color="Green" title="E65">A332</font></span></a>, <a class="ref" href="integra1.html#D4" onmouseover="rs('integra1/D4')" onmouseout="rh()">INTEGRA1:def 4</a>, <a class="txt" href="#E2:22_4_5_1_1_3"><span class="lab"><font color="Green" title="E85">A416</font></span></a></span>;<br/><a NAME="E3:22_4_5_1_1_3_1_1"/><span class="lab"><font color="Green" title="E96">A435</font></span>: 
<font color="Maroon" title="c31">i</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 <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="c16">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c26">n</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:22_4_5_1_1"><span class="lab"><font color="Green" title="E65">A332</font></span></a>, <a class="txt" href="#E1:22_4_5_1_1_3_1_1"><span class="lab"><font color="Green" title="E94">A429</font></span></a>, <a class="txt" href="#E2:22_4_5_1_1_3"><span class="lab"><font color="Green" title="E85">A416</font></span></a>, <a class="ref" href="integra1.html#T7" onmouseover="rs('integra1/T7')" onmouseout="rh()">INTEGRA1:7</a></span>;<br/><a NAME="E4:22_4_5_1_1_3_1_1"/><span class="lab"><font color="Green" title="E97">A440</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c16">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c26">n</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c36">k</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c9">a</font>,<font color="Maroon" title="c10">b</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:22"><span class="lab"><font color="Green" title="E28">A7</font></span></a>, <a class="txt" href="#E9:22_4_5_1_1_3"><span class="lab"><font color="Green" title="E88">A425</font></span></a>, <a class="ref" href="integra1.html#T6" onmouseover="rs('integra1/T6')" onmouseout="rh()">INTEGRA1:6</a></span>;<br/><a NAME="E5:22_4_5_1_1_3_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E98">A446</font></span>: 
( <font color="Maroon" title="c9">a</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c26">n</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c36">k</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c16">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c26">n</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c36">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c10">b</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_1.html#T1" onmouseover="rs('xxreal_1/T1')" onmouseout="rh()">XXREAL_1:1</a></span>;<br/><a NAME="E6:22_4_5_1_1_3_1_1"/><span class="lab"><font color="Green" title="E99">A445</font></span>: 
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c9">a</font>,<font color="Maroon" title="c10">b</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">A</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:22"><span class="lab"><font color="Green" title="E28">A7</font></span></a>, <a class="ref" href="funcop_1.html#T13" onmouseover="rs('funcop_1/T13')" onmouseout="rh()">FUNCOP_1:13</a></span>;<br/><span class="lab"><font color="Green" title="E100">A447</font></span>:  <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c9">a</font>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c16">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c26">n</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c36">k</font></span>)</span></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> 1</span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p2">(<span class="default"><span class="p3"><a href="rcomp_1.html#K4" title="RCOMP_1:func.4">].</a><span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c16">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c26">n</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c36">k</font></span>)</span>,<font color="Maroon" title="c10">b</font></span><a href="rcomp_1.html#K4" title="RCOMP_1:func.4">.]</a></span> <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p2">(<span class="default"><span class="p3"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c9">a</font>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c16">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c26">n</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c36">k</font></span>)</span></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> 1</span>)</span></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p2">(<span class="default"><span class="p3"><a href="rcomp_1.html#K4" title="RCOMP_1:func.4">].</a><span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c16">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c26">n</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c36">k</font></span>)</span>,<font color="Maroon" title="c10">b</font></span><a href="rcomp_1.html#K4" title="RCOMP_1:func.4">.]</a></span> <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_4.html#D1" onmouseover="rs('funct_4/D1')" onmouseout="rh()">FUNCT_4:def 1</a></span>
<br/>.= 
<font color="Maroon" title="c1">A</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:22"><span class="lab"><font color="Green" title="E28">A7</font></span></a>, <a class="txt" href="#E16:22_4_5_1_1_3"><span class="lab"><font color="Green" title="E93">B30</font></span></a>, <a class="txt" href="#E5:22_4_5_1_1_3_1_1"><span class="lab"><font color="Green" title="E98">A446</font></span></a>, <a class="ref" href="xxreal_1.html#T167" onmouseover="rs('xxreal_1/T167')" onmouseout="rh()">XXREAL_1:167</a></span>
;<br/><a NAME="E8:22_4_5_1_1_3_1_1"/>
 <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <span class="p1">(<span class="default"><span class="p2"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c9">a</font>,<font color="Maroon" title="c10">b</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> 
 ;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c46">f0</font> = <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c9">a</font>,<font color="Maroon" title="c10">b</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> as   <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> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:22_4_5_1_1_3_1_1"><span class="lab"><font color="Green" title="E99">A445</font></span></a>, <a class="ref" href="funct_2.html#T2" onmouseover="rs('funct_2/T2')" onmouseout="rh()">FUNCT_2:2</a></span>;<br/><a NAME="E10:22_4_5_1_1_3_1_1"/>
 <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c9">a</font>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c16">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c26">n</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c36">k</font></span>)</span></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> 1</span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p2">(<span class="default"><span class="p3"><a href="rcomp_1.html#K4" title="RCOMP_1:func.4">].</a><span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c16">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c26">n</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c36">k</font></span>)</span>,<font color="Maroon" title="c10">b</font></span><a href="rcomp_1.html#K4" title="RCOMP_1:func.4">.]</a></span> <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span>)</span></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> 
 ;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c47">f1</font> = <span class="p1">(<span class="default"><span class="p2"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c9">a</font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c16">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c26">n</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c36">k</font></span>)</span></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> 1</span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p1">(<span class="default"><span class="p2"><a href="rcomp_1.html#K4" title="RCOMP_1:func.4">].</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c16">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c26">n</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c36">k</font></span>)</span>,<font color="Maroon" title="c10">b</font></span><a href="rcomp_1.html#K4" title="RCOMP_1:func.4">.]</a></span> <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span>)</span> as   <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> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:22_4_5_1_1_3_1_1"><span class="lab"><font color="Green" title="E100">A447</font></span></a>, <a class="ref" href="funct_2.html#T2" onmouseover="rs('funct_2/T2')" onmouseout="rh()">FUNCT_2:2</a></span>;<br/><a NAME="E12:22_4_5_1_1_3_1_1"/><span class="lab"><font color="Green" title="E101">A451</font></span>: 
<font color="Maroon" title="c9">a</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c26">n</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c36">k</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:22_4"><span class="lab"><font color="Green" title="E42">A264</font></span></a>, <a class="txt" href="#E1:22_4_5_1_1_3_1_1"><span class="lab"><font color="Green" title="E94">A429</font></span></a></span>;<br/><span class="lab"><font color="Green" title="E102">A452</font></span>:  <a href="dualsp05.html#K2" title="DUALSP05:func.2">Dp1</a> (<font color="Maroon" title="c11">m</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c16">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c26">n</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c36">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>) = 
<font color="Maroon" title="c11">m</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c16">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c26">n</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c36">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:22_4_5_1_1_3"><span class="lab"><font color="Green" title="E92">A428</font></span></a>, <a class="ref" href="dualsp05.html#D2" target="_self" onmouseover="rs('dualsp05/D2')" onmouseout="rh()">defDp1</a>, <a class="txt" href="#E1:22_4_5_1_1_3_1_1"><span class="lab"><font color="Green" title="E94">A429</font></span></a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c9">a</font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c16">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c26">n</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c36">k</font></span>)</span></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> 1</span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p1">(<span class="default"><span class="p2"><a href="rcomp_1.html#K4" title="RCOMP_1:func.4">].</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c16">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c26">n</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c36">k</font></span>)</span>,<font color="Maroon" title="c10">b</font></span><a href="rcomp_1.html#K4" title="RCOMP_1:func.4">.]</a></span> <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:22"><span class="lab"><font color="Green" title="E29">A8</font></span></a>, <a class="txt" href="#E4:22_4_5_1_1_3_1_1"><span class="lab"><font color="Green" title="E97">A440</font></span></a>, <a class="txt" href="#E12:22_4_5_1_1_3_1_1"><span class="lab"><font color="Green" title="E101">A451</font></span></a></span>
;<br/><a NAME="E14:22_4_5_1_1_3_1_1"/>
<font color="Maroon" title="c1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><span class="p2">(<span class="default"><a href="seq_4.html#K3" title="SEQ_4:func.3">lower_bound</a> <font color="Maroon" title="c1">A</font></span>)</span>,<span class="p2">(<span class="default"><a href="seq_4.html#K2" title="SEQ_4:func.2">upper_bound</a> <font color="Maroon" title="c1">A</font></span>)</span></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="integra1.html#T4" onmouseover="rs('integra1/T4')" onmouseout="rh()">INTEGRA1:4</a></span>;<br/><a NAME="E15:22_4_5_1_1_3_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E103">A438</font></span>: 
 <a href="seq_4.html#K3" title="SEQ_4:func.3">lower_bound</a> <font color="Maroon" title="c1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">a</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:22"><span class="lab"><font color="Green" title="E28">A7</font></span></a>, <a class="ref" href="integra1.html#T5" onmouseover="rs('integra1/T5')" onmouseout="rh()">INTEGRA1:5</a></span>;<br/><a NAME="E16:22_4_5_1_1_3_1_1"/><span class="lab"><font color="Green" title="E104">A439</font></span>: 
<font color="Maroon" title="c9">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c9">a</font>,<font color="Maroon" title="c10">b</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:22"><span class="lab"><font color="Green" title="E28">A7</font></span></a></span>;<br/><span class="lab"><font color="Green" title="E105">A443</font></span>:  <a href="dualsp05.html#K2" title="DUALSP05:func.2">Dp1</a> (<font color="Maroon" title="c11">m</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c16">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c26">n</font></span>)</span>,<font color="Maroon" title="c36">k</font>) = 
<font color="Maroon" title="c11">m</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="seq_4.html#K3" title="SEQ_4:func.3">lower_bound</a> <font color="Maroon" title="c1">A</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:22_4_5_1_1_3"><span class="lab"><font color="Green" title="E90">A427</font></span></a>, <a class="txt" href="#E1:22_4_5_1_1_3_1_1"><span class="lab"><font color="Green" title="E94">A429</font></span></a>, <a class="ref" href="dualsp05.html#D2" target="_self" onmouseover="rs('dualsp05/D2')" onmouseout="rh()">defDp1</a></span>
<br/>.= 
<span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c9">a</font>,<font color="Maroon" title="c10">b</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:22"><span class="lab"><font color="Green" title="E29">A8</font></span></a>, <a class="txt" href="#E15:22_4_5_1_1_3_1_1"><span class="lab"><font color="Green" title="E103">A438</font></span></a>, <a class="txt" href="#E16:22_4_5_1_1_3_1_1"><span class="lab"><font color="Green" title="E104">A439</font></span></a></span>
;<br/><a NAME="E18:22_4_5_1_1_3_1_1"/><span class="lab"><font color="Green" title="E106">A453</font></span>: 
<font color="Maroon" title="c38">qk</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c30">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c37">r</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c47">f1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c30">t</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c46">f0</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c30">t</font></span>)</span></span>)</span>
 <div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="22_4_5_1_1_3_1_1_4"><span class="kw">proof </span></a><div class="add">

<span class="kw">reconsider </span><font color="Maroon" title="c48">H</font> = <span class="p1">(<span class="default"><a href="dualsp05.html#K2" title="DUALSP05:func.2">Dp1</a> (<font color="Maroon" title="c11">m</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c16">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c26">n</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c36">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>)</span>)</span> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <span class="p1">(<span class="default"><a href="dualsp05.html#K2" title="DUALSP05:func.2">Dp1</a> (<font color="Maroon" title="c11">m</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c16">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c26">n</font></span>)</span>,<font color="Maroon" title="c36">k</font>)</span>)</span> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="c0sp1.html#K11" title="C0SP1:func.11">R_Normed_Algebra_of_BoundedFunctions</a> <font color="Maroon" title="c1">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="c49">h</font> = <font color="Maroon" title="c48">H</font> as   <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> <span class="kw">by</span> <span class="lab"><a class="ref" href="dualsp05.html#T11" target="_self" onmouseover="rs('dualsp05/T11')" onmouseout="rh()">LM88</a></span>;<br/>
<a NAME="E3:22_4_5_1_1_3_1_1_4"/>
<font color="Maroon" title="c38">qk</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c37">r</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c48">H</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="dualsp05.html#T5" target="_self" onmouseover="rs('dualsp05/T5')" onmouseout="rh()">Lm1</a>, <a class="txt" href="#E8:22_4_5_1_1_3"><span class="lab"><font color="Green" title="E87">A423</font></span></a>, <a class="txt" href="#E6:22_4_5_1_1_3"><span class="lab"><font color="Green" title="E86">A420</font></span></a></span>;<br/>
<span class="kw">then </span><font color="Maroon" title="c38">qk</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c30">t</font> = 
<font color="Maroon" title="c37">r</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c49">h</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c30">t</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="c0sp1.html#T30" onmouseover="rs('c0sp1/T30')" onmouseout="rh()">C0SP1:30</a></span>
<br/>.= 
<font color="Maroon" title="c37">r</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c47">f1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c30">t</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c46">f0</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c30">t</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:22"><span class="lab"><font color="Green" title="E23">A2</font></span></a>, <a class="txt" href="#E17:22_4_5_1_1_3_1_1"><span class="lab"><font color="Green" title="E105">A443</font></span></a>, <a class="txt" href="#E13:22_4_5_1_1_3_1_1"><span class="lab"><font color="Green" title="E102">A452</font></span></a>, <a class="ref" href="c0sp1.html#T34" onmouseover="rs('c0sp1/T34')" onmouseout="rh()">C0SP1:34</a></span>
;<br/>
<span class="kw">hence </span><a NAME="E5:22_4_5_1_1_3_1_1_4"/>
<font color="Maroon" title="c38">qk</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c30">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c37">r</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c47">f1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c30">t</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c46">f0</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c30">t</font></span>)</span></span>)</span>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><a NAME="E19:22_4_5_1_1_3_1_1"/>
<font color="Maroon" title="c36">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c31">i</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:22_4_5_1_1_3_1_1"><span class="lab"><font color="Green" title="E94">A429</font></span></a>, <a class="txt" href="#E2:22_4_5_1_1_3"><span class="lab"><font color="Green" title="E85">A416</font></span></a>, <a class="txt" href="#E19:22_4_5_1_1"><span class="lab"><font color="Green" title="E80">A354</font></span></a>, <a class="ref" href="xxreal_0.html#T1" onmouseover="rs('xxreal_0/T1')" onmouseout="rh()">XXREAL_0:1</a></span>;<br/><a NAME="E20:22_4_5_1_1_3_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c36">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c31">i</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T13" onmouseover="rs('nat_1/T13')" onmouseout="rh()">NAT_1:13</a></span>;<br/><a NAME="E21:22_4_5_1_1_3_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E107">A456</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c36">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c31">i</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T13" onmouseover="rs('xreal_1/T13')" onmouseout="rh()">XREAL_1:13</a></span>;<br/><a NAME="E22:22_4_5_1_1_3_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c16">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c26">n</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c36">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c26">n</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c31">i</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span>
 <div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="22_4_5_1_1_3_1_1_5"><span class="kw">proof </span></a><div class="add">

<a NAME="E1:22_4_5_1_1_3_1_1_5"/>
( <font color="Maroon" title="c36">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c31">i</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 or <font color="Maroon" title="c36">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c31">i</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E21:22_4_5_1_1_3_1_1"><span class="lab"><font color="Green" title="E107">A456</font></span></a>, <a class="ref" href="xxreal_0.html#T1" onmouseover="rs('xxreal_0/T1')" onmouseout="rh()">XXREAL_0:1</a></span>;<br/>
<span class="kw">hence </span><a NAME="E2:22_4_5_1_1_3_1_1_5"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c16">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c26">n</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c36">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c26">n</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c31">i</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:22_4_5_1_1_3"><span class="lab"><font color="Green" title="E88">A425</font></span></a>, <a class="txt" href="#E3:22_4_5_1_1_3_1_1"><span class="lab"><font color="Green" title="E96">A435</font></span></a>, <a class="ref" href="valued_0.html#D13" onmouseover="rs('valued_0/D13')" onmouseout="rh()">VALUED_0:def 13</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="E23:22_4_5_1_1_3_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E108">A457</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c16">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c26">n</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c36">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c30">t</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:22_4_5_1_1"><span class="lab"><font color="Green" title="E67">A334</font></span></a>, <a class="txt" href="#E1:22_4_5_1_1_3_1_1"><span class="lab"><font color="Green" title="E94">A429</font></span></a>, <a class="txt" href="#E2:22_4_5_1_1_3"><span class="lab"><font color="Green" title="E85">A416</font></span></a>, <a class="ref" href="xxreal_0.html#T2" onmouseover="rs('xxreal_0/T2')" onmouseout="rh()">XXREAL_0:2</a>, <a class="txt" href="#E2:22_4_5_1_1_3_1_1"><span class="lab"><font color="Green" title="E95">A433</font></span></a></span>;<br/><a NAME="E24:22_4_5_1_1_3_1_1"/>
( <font color="Maroon" title="c9">a</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c30">t</font> &amp; <font color="Maroon" title="c30">t</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c10">b</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_1.html#T1" onmouseover="rs('xxreal_1/T1')" onmouseout="rh()">XXREAL_1:1</a>, <a class="txt" href="#E11:22"><span class="lab"><font color="Green" title="E28">A7</font></span></a></span>;<br/><a NAME="E25:22_4_5_1_1_3_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E109">A459</font></span>: 
<font color="Maroon" title="c30">t</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"><a href="rcomp_1.html#K4" title="RCOMP_1:func.4">].</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c16">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c26">n</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c36">k</font></span>)</span>,<font color="Maroon" title="c10">b</font></span><a href="rcomp_1.html#K4" title="RCOMP_1:func.4">.]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E23:22_4_5_1_1_3_1_1"><span class="lab"><font color="Green" title="E108">A457</font></span></a></span>;<br/><span class="kw">then </span><span class="lab"><font color="Green" title="E110">A461</font></span>: <font color="Maroon" title="c47">f1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c30">t</font> = 
<span class="p1">(<span class="default"><span class="p2"><a href="rcomp_1.html#K4" title="RCOMP_1:func.4">].</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c16">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c26">n</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c36">k</font></span>)</span>,<font color="Maroon" title="c10">b</font></span><a href="rcomp_1.html#K4" title="RCOMP_1:func.4">.]</a></span> <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c30">t</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:22_4_5_1_1_3"><span class="lab"><font color="Green" title="E93">B30</font></span></a>, <a class="ref" href="funct_4.html#T13" onmouseover="rs('funct_4/T13')" onmouseout="rh()">FUNCT_4:13</a></span>
<br/>.= 
 <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E25:22_4_5_1_1_3_1_1"><span class="lab"><font color="Green" title="E109">A459</font></span></a>, <a class="ref" href="funcop_1.html#T7" onmouseover="rs('funcop_1/T7')" onmouseout="rh()">FUNCOP_1:7</a></span>
;<br/><a NAME="E27:22_4_5_1_1_3_1_1"/>
<font color="Maroon" title="c46">f0</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c30">t</font> <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="txt" href="#E11:22"><span class="lab"><font color="Green" title="E28">A7</font></span></a>, <a class="ref" href="funcop_1.html#T7" onmouseover="rs('funcop_1/T7')" onmouseout="rh()">FUNCOP_1:7</a></span>;<br/><span class="kw">hence </span><a NAME="E28:22_4_5_1_1_3_1_1"/>
<font color="Maroon" title="c33">z</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c36">k</font> <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="txt" href="#E8:22_4_5_1_1_3"><span class="lab"><font color="Green" title="E87">A423</font></span></a>, <a class="txt" href="#E18:22_4_5_1_1_3_1_1"><span class="lab"><font color="Green" title="E106">A453</font></span></a>, <a class="txt" href="#E26:22_4_5_1_1_3_1_1"><span class="lab"><font color="Green" title="E110">A461</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><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="22_4_5_1_1_3_1_2"><span class="kw">suppose </span></a><a NAME="E1:22_4_5_1_1_3_1_2"/><span class="lab"><font color="Green" title="E94">A462</font></span>: 
<font color="Maroon" title="c36">k</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 1
 ; <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="c33">z</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c36">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> </span><br/><div class="add"><a NAME="E2:22_4_5_1_1_3_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E95">A464</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c16">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c26">n</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c36">k</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c9">a</font>,<font color="Maroon" title="c10">b</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:22"><span class="lab"><font color="Green" title="E28">A7</font></span></a>, <a class="txt" href="#E9:22_4_5_1_1_3"><span class="lab"><font color="Green" title="E88">A425</font></span></a>, <a class="ref" href="integra1.html#T7" onmouseover="rs('integra1/T7')" onmouseout="rh()">INTEGRA1:7</a></span>;<br/><a NAME="E3:22_4_5_1_1_3_1_2"/><span class="lab"><font color="Green" title="E96">A463</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c16">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c26">n</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c36">k</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c9">a</font>,<font color="Maroon" title="c10">b</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:22"><span class="lab"><font color="Green" title="E28">A7</font></span></a>, <a class="txt" href="#E9:22_4_5_1_1_3"><span class="lab"><font color="Green" title="E88">A425</font></span></a>, <a class="ref" href="integra1.html#T6" onmouseover="rs('integra1/T6')" onmouseout="rh()">INTEGRA1:6</a></span>;<br/><a NAME="E4:22_4_5_1_1_3_1_2"/><span class="lab"><font color="Green" title="E97">A467</font></span>: 
<font color="Maroon" title="c36">k</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 <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="c16">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c26">n</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:22_4_5_1_1_3"><span class="lab"><font color="Green" title="E88">A425</font></span></a>, <a class="txt" href="#E1:22_4_5_1_1_3_1_2"><span class="lab"><font color="Green" title="E94">A462</font></span></a>, <a class="ref" href="integra1.html#T7" onmouseover="rs('integra1/T7')" onmouseout="rh()">INTEGRA1:7</a></span>;<br/><a NAME="E5:22_4_5_1_1_3_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E98">A468</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c16">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c26">n</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c36">k</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c26">n</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c36">k</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:22_4_5_1_1_3"><span class="lab"><font color="Green" title="E88">A425</font></span></a>, <a class="ref" href="xreal_1.html#T146" onmouseover="rs('xreal_1/T146')" onmouseout="rh()">XREAL_1:146</a>, <a class="ref" href="valued_0.html#D13" onmouseover="rs('valued_0/D13')" onmouseout="rh()">VALUED_0:def 13</a></span>;<br/><a NAME="E6:22_4_5_1_1_3_1_2"/>
1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c36">k</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:22_4_5_1_1_3"><span class="lab"><font color="Green" title="E89">A426</font></span></a>, <a class="txt" href="#E1:22_4_5_1_1_3_1_2"><span class="lab"><font color="Green" title="E94">A462</font></span></a>, <a class="ref" href="xxreal_0.html#T1" onmouseover="rs('xxreal_0/T1')" onmouseout="rh()">XXREAL_0:1</a></span>;<br/><a NAME="E7:22_4_5_1_1_3_1_2"/><span class="kw">then </span>
1 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c36">k</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T13" onmouseover="rs('nat_1/T13')" onmouseout="rh()">NAT_1:13</a></span>;<br/><a NAME="E8:22_4_5_1_1_3_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E99">A470</font></span>: 
2 <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c36">k</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T13" onmouseover="rs('xreal_1/T13')" onmouseout="rh()">XREAL_1:13</a></span>;<br/><a NAME="E9:22_4_5_1_1_3_1_2"/>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c26">n</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:22_4_5_1_1_3"><span class="lab"><font color="Green" title="E89">A426</font></span></a>, <a class="ref" href="xxreal_0.html#T2" onmouseover="rs('xxreal_0/T2')" onmouseout="rh()">XXREAL_0:2</a></span>;<br/><a NAME="E10:22_4_5_1_1_3_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E100">A471</font></span>: 
1 <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="c16">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c26">n</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_3.html#T25" onmouseover="rs('finseq_3/T25')" onmouseout="rh()">FINSEQ_3:25</a></span>;<br/><a NAME="E11:22_4_5_1_1_3_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c16">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c26">n</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c26">n</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c36">k</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span>
 <div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="22_4_5_1_1_3_1_2_1"><span class="kw">proof </span></a><div class="add">

<a NAME="E1:22_4_5_1_1_3_1_2_1"/>
( 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c36">k</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 or 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c36">k</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:22_4_5_1_1_3_1_2"><span class="lab"><font color="Green" title="E99">A470</font></span></a>, <a class="ref" href="xxreal_0.html#T1" onmouseover="rs('xxreal_0/T1')" onmouseout="rh()">XXREAL_0:1</a></span>;<br/>
<span class="kw">hence </span><a NAME="E2:22_4_5_1_1_3_1_2_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c16">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c26">n</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c26">n</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c36">k</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:22_4_5_1_1_3_1_2"><span class="lab"><font color="Green" title="E97">A467</font></span></a>, <a class="txt" href="#E10:22_4_5_1_1_3_1_2"><span class="lab"><font color="Green" title="E100">A471</font></span></a>, <a class="ref" href="valued_0.html#D13" onmouseover="rs('valued_0/D13')" onmouseout="rh()">VALUED_0:def 13</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="E12:22_4_5_1_1_3_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E101">A472</font></span>: 
<font color="Maroon" title="c9">a</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c26">n</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c36">k</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:22_4"><span class="lab"><font color="Green" title="E42">A264</font></span></a>, <a class="ref" href="xxreal_0.html#T2" onmouseover="rs('xxreal_0/T2')" onmouseout="rh()">XXREAL_0:2</a></span>;<br/><a NAME="E13:22_4_5_1_1_3_1_2"/>
1 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c36">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T8" onmouseover="rs('xreal_1/T8')" onmouseout="rh()">XREAL_1:8</a>, <a class="txt" href="#E10:22_4_5_1_1_3"><span class="lab"><font color="Green" title="E89">A426</font></span></a></span>;<br/><span class="kw">then </span><span class="lab"><font color="Green" title="E102">A474</font></span>:  <a href="dualsp05.html#K2" title="DUALSP05:func.2">Dp1</a> (<font color="Maroon" title="c11">m</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c16">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c26">n</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c36">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>) = 
<font color="Maroon" title="c11">m</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c16">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c26">n</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c36">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:22_4_5_1_1_3"><span class="lab"><font color="Green" title="E92">A428</font></span></a>, <a class="ref" href="dualsp05.html#D2" target="_self" onmouseover="rs('dualsp05/D2')" onmouseout="rh()">defDp1</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c9">a</font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c16">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c26">n</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c36">k</font></span>)</span></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> 1</span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p1">(<span class="default"><span class="p2"><a href="rcomp_1.html#K4" title="RCOMP_1:func.4">].</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c16">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c26">n</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c36">k</font></span>)</span>,<font color="Maroon" title="c10">b</font></span><a href="rcomp_1.html#K4" title="RCOMP_1:func.4">.]</a></span> <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:22"><span class="lab"><font color="Green" title="E29">A8</font></span></a>, <a class="txt" href="#E3:22_4_5_1_1_3_1_2"><span class="lab"><font color="Green" title="E96">A463</font></span></a>, <a class="txt" href="#E12:22_4_5_1_1_3_1_2"><span class="lab"><font color="Green" title="E101">A472</font></span></a>, <a class="txt" href="#E5:22_4_5_1_1_3_1_2"><span class="lab"><font color="Green" title="E98">A468</font></span></a></span>
;<br/><span class="lab"><font color="Green" title="E103">A475</font></span>:  <a href="dualsp05.html#K2" title="DUALSP05:func.2">Dp1</a> (<font color="Maroon" title="c11">m</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c16">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c26">n</font></span>)</span>,<font color="Maroon" title="c36">k</font>) = 
<font color="Maroon" title="c11">m</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c16">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c26">n</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c36">k</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:22_4_5_1_1_3"><span class="lab"><font color="Green" title="E90">A427</font></span></a>, <a class="txt" href="#E1:22_4_5_1_1_3_1_2"><span class="lab"><font color="Green" title="E94">A462</font></span></a>, <a class="ref" href="dualsp05.html#D2" target="_self" onmouseover="rs('dualsp05/D2')" onmouseout="rh()">defDp1</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c9">a</font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c16">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c26">n</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c36">k</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span></span>)</span></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> 1</span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p1">(<span class="default"><span class="p2"><a href="rcomp_1.html#K4" title="RCOMP_1:func.4">].</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c16">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c26">n</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c36">k</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c10">b</font></span><a href="rcomp_1.html#K4" title="RCOMP_1:func.4">.]</a></span> <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:22"><span class="lab"><font color="Green" title="E29">A8</font></span></a>, <a class="txt" href="#E2:22_4_5_1_1_3_1_2"><span class="lab"><font color="Green" title="E95">A464</font></span></a>, <a class="txt" href="#E12:22_4_5_1_1_3_1_2"><span class="lab"><font color="Green" title="E101">A472</font></span></a></span>
;<br/><a NAME="E16:22_4_5_1_1_3_1_2"/><span class="lab"><font color="Green" title="E104">A477</font></span>: 
( <font color="Maroon" title="c9">a</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c26">n</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c36">k</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c16">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c26">n</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c36">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c10">b</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:22_4_5_1_1_3_1_2"><span class="lab"><font color="Green" title="E96">A463</font></span></a>, <a class="ref" href="xxreal_1.html#T1" onmouseover="rs('xxreal_1/T1')" onmouseout="rh()">XXREAL_1:1</a></span>;<br/><a NAME="E17:22_4_5_1_1_3_1_2"/><span class="lab"><font color="Green" title="E105">A478</font></span>: 
( <font color="Maroon" title="c9">a</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c26">n</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c36">k</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c16">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c26">n</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c36">k</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c10">b</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:22_4_5_1_1_3_1_2"><span class="lab"><font color="Green" title="E95">A464</font></span></a>, <a class="ref" href="xxreal_1.html#T1" onmouseover="rs('xxreal_1/T1')" onmouseout="rh()">XXREAL_1:1</a></span>;<br/><span class="lab"><font color="Green" title="E106">A479</font></span>:  <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c9">a</font>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c16">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c26">n</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c36">k</font></span>)</span></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> 1</span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p2">(<span class="default"><span class="p3"><a href="rcomp_1.html#K4" title="RCOMP_1:func.4">].</a><span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c16">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c26">n</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c36">k</font></span>)</span>,<font color="Maroon" title="c10">b</font></span><a href="rcomp_1.html#K4" title="RCOMP_1:func.4">.]</a></span> <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p2">(<span class="default"><span class="p3"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c9">a</font>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c16">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c26">n</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c36">k</font></span>)</span></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> 1</span>)</span></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p2">(<span class="default"><span class="p3"><a href="rcomp_1.html#K4" title="RCOMP_1:func.4">].</a><span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c16">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c26">n</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c36">k</font></span>)</span>,<font color="Maroon" title="c10">b</font></span><a href="rcomp_1.html#K4" title="RCOMP_1:func.4">.]</a></span> <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_4.html#D1" onmouseover="rs('funct_4/D1')" onmouseout="rh()">FUNCT_4:def 1</a></span>
<br/>.= 
<font color="Maroon" title="c1">A</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:22"><span class="lab"><font color="Green" title="E28">A7</font></span></a>, <a class="txt" href="#E16:22_4_5_1_1_3"><span class="lab"><font color="Green" title="E93">B30</font></span></a>, <a class="txt" href="#E16:22_4_5_1_1_3_1_2"><span class="lab"><font color="Green" title="E104">A477</font></span></a>, <a class="ref" href="xxreal_1.html#T167" onmouseover="rs('xxreal_1/T167')" onmouseout="rh()">XXREAL_1:167</a></span>
;<br/><span class="lab"><font color="Green" title="E107">A480</font></span>:  <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c9">a</font>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c16">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c26">n</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p5">(<span class="default"><font color="Maroon" title="c36">k</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span></span>)</span></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> 1</span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p2">(<span class="default"><span class="p3"><a href="rcomp_1.html#K4" title="RCOMP_1:func.4">].</a><span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c16">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c26">n</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p5">(<span class="default"><font color="Maroon" title="c36">k</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c10">b</font></span><a href="rcomp_1.html#K4" title="RCOMP_1:func.4">.]</a></span> <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p2">(<span class="default"><span class="p3"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c9">a</font>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c16">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c26">n</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p5">(<span class="default"><font color="Maroon" title="c36">k</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span></span>)</span></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> 1</span>)</span></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p2">(<span class="default"><span class="p3"><a href="rcomp_1.html#K4" title="RCOMP_1:func.4">].</a><span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c16">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c26">n</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p5">(<span class="default"><font color="Maroon" title="c36">k</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c10">b</font></span><a href="rcomp_1.html#K4" title="RCOMP_1:func.4">.]</a></span> <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_4.html#D1" onmouseover="rs('funct_4/D1')" onmouseout="rh()">FUNCT_4:def 1</a></span>
<br/>.= 
<font color="Maroon" title="c1">A</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:22"><span class="lab"><font color="Green" title="E28">A7</font></span></a>, <a class="txt" href="#E16:22_4_5_1_1_3"><span class="lab"><font color="Green" title="E93">B30</font></span></a>, <a class="txt" href="#E17:22_4_5_1_1_3_1_2"><span class="lab"><font color="Green" title="E105">A478</font></span></a>, <a class="ref" href="xxreal_1.html#T167" onmouseover="rs('xxreal_1/T167')" onmouseout="rh()">XXREAL_1:167</a></span>
;<br/><a NAME="E20:22_4_5_1_1_3_1_2"/>
 <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c9">a</font>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c16">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c26">n</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c36">k</font></span>)</span></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> 1</span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p2">(<span class="default"><span class="p3"><a href="rcomp_1.html#K4" title="RCOMP_1:func.4">].</a><span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c16">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c26">n</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c36">k</font></span>)</span>,<font color="Maroon" title="c10">b</font></span><a href="rcomp_1.html#K4" title="RCOMP_1:func.4">.]</a></span> <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span>)</span></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> 
 ;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c46">f1</font> = <span class="p1">(<span class="default"><span class="p2"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c9">a</font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c16">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c26">n</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c36">k</font></span>)</span></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> 1</span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p1">(<span class="default"><span class="p2"><a href="rcomp_1.html#K4" title="RCOMP_1:func.4">].</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c16">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c26">n</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c36">k</font></span>)</span>,<font color="Maroon" title="c10">b</font></span><a href="rcomp_1.html#K4" title="RCOMP_1:func.4">.]</a></span> <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span>)</span> as   <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> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:22_4_5_1_1_3_1_2"><span class="lab"><font color="Green" title="E106">A479</font></span></a>, <a class="ref" href="funct_2.html#T2" onmouseover="rs('funct_2/T2')" onmouseout="rh()">FUNCT_2:2</a></span>;<br/><a NAME="E22:22_4_5_1_1_3_1_2"/>
 <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c9">a</font>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c16">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c26">n</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p5">(<span class="default"><font color="Maroon" title="c36">k</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span></span>)</span></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> 1</span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p2">(<span class="default"><span class="p3"><a href="rcomp_1.html#K4" title="RCOMP_1:func.4">].</a><span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c16">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c26">n</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p5">(<span class="default"><font color="Maroon" title="c36">k</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c10">b</font></span><a href="rcomp_1.html#K4" title="RCOMP_1:func.4">.]</a></span> <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span>)</span></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> 
 ;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c47">f2</font> = <span class="p1">(<span class="default"><span class="p2"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c9">a</font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c16">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c26">n</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c36">k</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span></span>)</span></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> 1</span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p1">(<span class="default"><span class="p2"><a href="rcomp_1.html#K4" title="RCOMP_1:func.4">].</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c16">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c26">n</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c36">k</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c10">b</font></span><a href="rcomp_1.html#K4" title="RCOMP_1:func.4">.]</a></span> <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span>)</span> as   <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> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:22_4_5_1_1_3_1_2"><span class="lab"><font color="Green" title="E107">A480</font></span></a>, <a class="ref" href="funct_2.html#T2" onmouseover="rs('funct_2/T2')" onmouseout="rh()">FUNCT_2:2</a></span>;<br/><a NAME="E24:22_4_5_1_1_3_1_2"/><span class="lab"><font color="Green" title="E108">A482</font></span>: 
<font color="Maroon" title="c38">qk</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c30">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c37">r</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c46">f1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c30">t</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c47">f2</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c30">t</font></span>)</span></span>)</span>
 <div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="22_4_5_1_1_3_1_2_6"><span class="kw">proof </span></a><div class="add">

<span class="kw">reconsider </span><font color="Maroon" title="c48">H</font> = <span class="p1">(<span class="default"><a href="dualsp05.html#K2" title="DUALSP05:func.2">Dp1</a> (<font color="Maroon" title="c11">m</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c16">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c26">n</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c36">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>)</span>)</span> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <span class="p1">(<span class="default"><a href="dualsp05.html#K2" title="DUALSP05:func.2">Dp1</a> (<font color="Maroon" title="c11">m</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c16">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c26">n</font></span>)</span>,<font color="Maroon" title="c36">k</font>)</span>)</span> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="c0sp1.html#K11" title="C0SP1:func.11">R_Normed_Algebra_of_BoundedFunctions</a> <font color="Maroon" title="c1">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="c49">h</font> = <font color="Maroon" title="c48">H</font> as   <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> <span class="kw">by</span> <span class="lab"><a class="ref" href="dualsp05.html#T11" target="_self" onmouseover="rs('dualsp05/T11')" onmouseout="rh()">LM88</a></span>;<br/>
<a NAME="E3:22_4_5_1_1_3_1_2_6"/>
<font color="Maroon" title="c38">qk</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c37">r</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c48">H</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="dualsp05.html#T5" target="_self" onmouseover="rs('dualsp05/T5')" onmouseout="rh()">Lm1</a>, <a class="txt" href="#E8:22_4_5_1_1_3"><span class="lab"><font color="Green" title="E87">A423</font></span></a>, <a class="txt" href="#E6:22_4_5_1_1_3"><span class="lab"><font color="Green" title="E86">A420</font></span></a></span>;<br/>
<span class="kw">then </span><font color="Maroon" title="c38">qk</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c30">t</font> = 
<font color="Maroon" title="c37">r</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c49">h</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c30">t</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="c0sp1.html#T30" onmouseover="rs('c0sp1/T30')" onmouseout="rh()">C0SP1:30</a></span>
<br/>.= 
<font color="Maroon" title="c37">r</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c46">f1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c30">t</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c47">f2</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c30">t</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:22"><span class="lab"><font color="Green" title="E23">A2</font></span></a>, <a class="txt" href="#E14:22_4_5_1_1_3_1_2"><span class="lab"><font color="Green" title="E102">A474</font></span></a>, <a class="txt" href="#E15:22_4_5_1_1_3_1_2"><span class="lab"><font color="Green" title="E103">A475</font></span></a>, <a class="ref" href="c0sp1.html#T34" onmouseover="rs('c0sp1/T34')" onmouseout="rh()">C0SP1:34</a></span>
;<br/>
<span class="kw">hence </span><a NAME="E5:22_4_5_1_1_3_1_2_6"/>
<font color="Maroon" title="c38">qk</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c30">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c37">r</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c46">f1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c30">t</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c47">f2</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c30">t</font></span>)</span></span>)</span>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="22_4_5_1_1_3_1_2_7"><span class="kw">per </span></a><a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:22_4_5_1_1_3_1_2_7"/>
( <font color="Maroon" title="c31">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c36">k</font> or <font color="Maroon" title="c36">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c31">i</font> )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:22_4_5_1_1_3"><span class="lab"><font color="Green" title="E85">A416</font></span></a>, <a class="ref" href="xxreal_0.html#T1" onmouseover="rs('xxreal_0/T1')" onmouseout="rh()">XXREAL_0:1</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="22_4_5_1_1_3_1_2_7_1"><span class="kw">suppose </span></a><a NAME="E1:22_4_5_1_1_3_1_2_7_1"/>
<font color="Maroon" title="c31">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c36">k</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="c33">z</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c36">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> </span><br/><div class="add"><a NAME="E2:22_4_5_1_1_3_1_2_7_1"/><span class="kw">then </span>
<font color="Maroon" title="c31">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c36">k</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T13" onmouseover="rs('nat_1/T13')" onmouseout="rh()">NAT_1:13</a></span>;<br/><a NAME="E3:22_4_5_1_1_3_1_2_7_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E109">A485</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c31">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c36">k</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T13" onmouseover="rs('xreal_1/T13')" onmouseout="rh()">XREAL_1:13</a></span>;<br/><a NAME="E4:22_4_5_1_1_3_1_2_7_1"/><span class="lab"><font color="Green" title="E110">A486</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c16">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c26">n</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c31">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c26">n</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c36">k</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span>
 <div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="22_4_5_1_1_3_1_2_7_1_1"><span class="kw">proof </span></a><div class="add">

<a NAME="E1:22_4_5_1_1_3_1_2_7_1_1"/>
( <font color="Maroon" title="c31">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c36">k</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 or <font color="Maroon" title="c31">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c36">k</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:22_4_5_1_1_3_1_2_7_1"><span class="lab"><font color="Green" title="E109">A485</font></span></a>, <a class="ref" href="xxreal_0.html#T1" onmouseover="rs('xxreal_0/T1')" onmouseout="rh()">XXREAL_0:1</a></span>;<br/>
<span class="kw">hence </span><a NAME="E2:22_4_5_1_1_3_1_2_7_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c16">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c26">n</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c31">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c26">n</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c36">k</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:22_4_5_1_1"><span class="lab"><font color="Green" title="E65">A332</font></span></a>, <a class="txt" href="#E4:22_4_5_1_1_3_1_2"><span class="lab"><font color="Green" title="E97">A467</font></span></a>, <a class="ref" href="valued_0.html#D13" onmouseover="rs('valued_0/D13')" onmouseout="rh()">VALUED_0:def 13</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="E5:22_4_5_1_1_3_1_2_7_1"/><span class="lab"><font color="Green" title="E111">A488</font></span>: 
 <a href="seq_4.html#K2" title="SEQ_4:func.2">upper_bound</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c16">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c26">n</font></span>)</span>,<font color="Maroon" title="c31">i</font>)</span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c26">n</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c36">k</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span>
 <div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="22_4_5_1_1_3_1_2_7_1_2"><span class="kw">proof </span></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="22_4_5_1_1_3_1_2_7_1_2_1"><span class="kw">per </span></a><a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:22_4_5_1_1_3_1_2_7_1_2_1"/>
( <font color="Maroon" title="c31">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or <font color="Maroon" title="c31">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 1 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="22_4_5_1_1_3_1_2_7_1_2_1_1"><span class="kw">suppose </span></a><a NAME="E1:22_4_5_1_1_3_1_2_7_1_2_1_1"/>
<font color="Maroon" title="c31">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 ; <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="seq_4.html#K2" title="SEQ_4:func.2">upper_bound</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c16">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c26">n</font></span>)</span>,<font color="Maroon" title="c31">i</font>)</span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c26">n</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c36">k</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span></span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:22_4_5_1_1_3_1_2_7_1_2_1_1"/>
 <a href="seq_4.html#K2" title="SEQ_4:func.2">upper_bound</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c16">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c26">n</font></span>)</span>,<font color="Maroon" title="c31">i</font>)</span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c26">n</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c36">k</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:22_4_5_1_1_3_1_2_7_1"><span class="lab"><font color="Green" title="E110">A486</font></span></a>, <a class="txt" href="#E1:22_4_5_1_1"><span class="lab"><font color="Green" title="E65">A332</font></span></a>, <a class="ref" href="integra1.html#D4" onmouseover="rs('integra1/D4')" onmouseout="rh()">INTEGRA1:def 4</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><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="22_4_5_1_1_3_1_2_7_1_2_1_2"><span class="kw">suppose </span></a><a NAME="E1:22_4_5_1_1_3_1_2_7_1_2_1_2"/>
<font color="Maroon" title="c31">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 1
 ; <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="seq_4.html#K2" title="SEQ_4:func.2">upper_bound</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c16">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c26">n</font></span>)</span>,<font color="Maroon" title="c31">i</font>)</span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c26">n</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c36">k</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span></span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:22_4_5_1_1_3_1_2_7_1_2_1_2"/>
 <a href="seq_4.html#K2" title="SEQ_4:func.2">upper_bound</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c16">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c26">n</font></span>)</span>,<font color="Maroon" title="c31">i</font>)</span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c26">n</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c36">k</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:22_4_5_1_1_3_1_2_7_1"><span class="lab"><font color="Green" title="E110">A486</font></span></a>, <a class="txt" href="#E1:22_4_5_1_1"><span class="lab"><font color="Green" title="E65">A332</font></span></a>, <a class="ref" href="integra1.html#D4" onmouseover="rs('integra1/D4')" onmouseout="rh()">INTEGRA1:def 4</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></div><span class="kw">end;</span></div>

</div><span class="kw">end;</span></div><a NAME="E6:22_4_5_1_1_3_1_2_7_1"/>
( <font color="Maroon" title="c9">a</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c30">t</font> &amp; <font color="Maroon" title="c30">t</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="seq_4.html#K2" title="SEQ_4:func.2">upper_bound</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c16">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c26">n</font></span>)</span>,<font color="Maroon" title="c31">i</font>)</span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:22_4_5_1_1"><span class="lab"><font color="Green" title="E79">B352</font></span></a>, <a class="ref" href="xxreal_1.html#T1" onmouseover="rs('xxreal_1/T1')" onmouseout="rh()">XXREAL_1:1</a>, <a class="txt" href="#E11:22"><span class="lab"><font color="Green" title="E28">A7</font></span></a></span>;<br/><a NAME="E7:22_4_5_1_1_3_1_2_7_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E112">A489</font></span>: 
( <font color="Maroon" title="c9">a</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c30">t</font> &amp; <font color="Maroon" title="c30">t</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c26">n</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c36">k</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:22_4_5_1_1_3_1_2_7_1"><span class="lab"><font color="Green" title="E111">A488</font></span></a>, <a class="ref" href="xxreal_0.html#T2" onmouseover="rs('xxreal_0/T2')" onmouseout="rh()">XXREAL_0:2</a></span>;<br/><a NAME="E8:22_4_5_1_1_3_1_2_7_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E113">A490</font></span>: 
<font color="Maroon" title="c30">t</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c9">a</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c16">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c26">n</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c36">k</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span></span>)</span></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span>
 ;<br/><a NAME="E9:22_4_5_1_1_3_1_2_7_1"/>
( <font color="Maroon" title="c9">a</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c30">t</font> &amp; <font color="Maroon" title="c30">t</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c26">n</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c36">k</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:22_4_5_1_1_3_1_2"><span class="lab"><font color="Green" title="E98">A468</font></span></a>, <a class="txt" href="#E7:22_4_5_1_1_3_1_2_7_1"><span class="lab"><font color="Green" title="E112">A489</font></span></a>, <a class="ref" href="xxreal_0.html#T2" onmouseover="rs('xxreal_0/T2')" onmouseout="rh()">XXREAL_0:2</a></span>;<br/><a NAME="E10:22_4_5_1_1_3_1_2_7_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E114">A491</font></span>: 
<font color="Maroon" title="c30">t</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c9">a</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c16">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c26">n</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c36">k</font></span>)</span></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span>
 ;<br/><span class="kw">then </span><span class="lab"><font color="Green" title="E115">A494</font></span>: <font color="Maroon" title="c46">f1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c30">t</font> = 
<span class="p1">(<span class="default"><span class="p2"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c9">a</font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c16">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c26">n</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c36">k</font></span>)</span></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> 1</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c30">t</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_4.html#T16" onmouseover="rs('funct_4/T16')" onmouseout="rh()">FUNCT_4:16</a>, <a class="txt" href="#E16:22_4_5_1_1_3"><span class="lab"><font color="Green" title="E93">B30</font></span></a>, <a class="ref" href="xxreal_1.html#T90" onmouseover="rs('xxreal_1/T90')" onmouseout="rh()">XXREAL_1:90</a></span>
<br/>.= 
1
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:22_4_5_1_1_3_1_2_7_1"><span class="lab"><font color="Green" title="E114">A491</font></span></a>, <a class="ref" href="funcop_1.html#T7" onmouseover="rs('funcop_1/T7')" onmouseout="rh()">FUNCOP_1:7</a></span>
;<br/><font color="Maroon" title="c47">f2</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c30">t</font> = 
<span class="p1">(<span class="default"><span class="p2"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c9">a</font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c16">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c26">n</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c36">k</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span></span>)</span></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> 1</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c30">t</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:22_4_5_1_1_3_1_2_7_1"><span class="lab"><font color="Green" title="E113">A490</font></span></a>, <a class="ref" href="funct_4.html#T16" onmouseover="rs('funct_4/T16')" onmouseout="rh()">FUNCT_4:16</a>, <a class="txt" href="#E16:22_4_5_1_1_3"><span class="lab"><font color="Green" title="E93">B30</font></span></a>, <a class="ref" href="xxreal_1.html#T90" onmouseover="rs('xxreal_1/T90')" onmouseout="rh()">XXREAL_1:90</a></span>
<br/>.= 
1
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:22_4_5_1_1_3_1_2_7_1"><span class="lab"><font color="Green" title="E113">A490</font></span></a>, <a class="ref" href="funcop_1.html#T7" onmouseover="rs('funcop_1/T7')" onmouseout="rh()">FUNCOP_1:7</a></span>
;<br/><span class="kw">hence </span><a NAME="E13:22_4_5_1_1_3_1_2_7_1"/>
<font color="Maroon" title="c33">z</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c36">k</font> <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="txt" href="#E8:22_4_5_1_1_3"><span class="lab"><font color="Green" title="E87">A423</font></span></a>, <a class="txt" href="#E24:22_4_5_1_1_3_1_2"><span class="lab"><font color="Green" title="E108">A482</font></span></a>, <a class="txt" href="#E11:22_4_5_1_1_3_1_2_7_1"><span class="lab"><font color="Green" title="E115">A494</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><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="22_4_5_1_1_3_1_2_7_2"><span class="kw">suppose </span></a><a NAME="E1:22_4_5_1_1_3_1_2_7_2"/><span class="lab"><font color="Green" title="E109">A496</font></span>: 
<font color="Maroon" title="c36">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c31">i</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="c33">z</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c36">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> </span><br/><div class="add"><a NAME="E2:22_4_5_1_1_3_1_2_7_2"/><span class="kw">then </span>
<font color="Maroon" title="c36">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c31">i</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T13" onmouseover="rs('nat_1/T13')" onmouseout="rh()">NAT_1:13</a></span>;<br/><a NAME="E3:22_4_5_1_1_3_1_2_7_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E110">A499</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c36">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c31">i</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T13" onmouseover="rs('xreal_1/T13')" onmouseout="rh()">XREAL_1:13</a></span>;<br/><a NAME="E4:22_4_5_1_1_3_1_2_7_2"/><span class="lab"><font color="Green" title="E111">A500</font></span>: 
<font color="Maroon" title="c31">i</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 <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="c16">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c26">n</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:22_4_5_1_1"><span class="lab"><font color="Green" title="E65">A332</font></span></a>, <a class="txt" href="#E1:22_4_5_1_1_3_1_2_7_2"><span class="lab"><font color="Green" title="E109">A496</font></span></a>, <a class="txt" href="#E10:22_4_5_1_1_3"><span class="lab"><font color="Green" title="E89">A426</font></span></a>, <a class="ref" href="integra1.html#T7" onmouseover="rs('integra1/T7')" onmouseout="rh()">INTEGRA1:7</a></span>;<br/><a NAME="E5:22_4_5_1_1_3_1_2_7_2"/><span class="lab"><font color="Green" title="E112">A502</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c16">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c26">n</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c36">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c26">n</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c31">i</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span>
 <div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="22_4_5_1_1_3_1_2_7_2_1"><span class="kw">proof </span></a><div class="add">

<a NAME="E1:22_4_5_1_1_3_1_2_7_2_1"/>
( <font color="Maroon" title="c36">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c31">i</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 or <font color="Maroon" title="c36">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c31">i</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:22_4_5_1_1_3_1_2_7_2"><span class="lab"><font color="Green" title="E110">A499</font></span></a>, <a class="ref" href="xxreal_0.html#T1" onmouseover="rs('xxreal_0/T1')" onmouseout="rh()">XXREAL_0:1</a></span>;<br/>
<span class="kw">hence </span><a NAME="E2:22_4_5_1_1_3_1_2_7_2_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c16">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c26">n</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c36">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c26">n</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c31">i</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:22_4_5_1_1_3"><span class="lab"><font color="Green" title="E88">A425</font></span></a>, <a class="txt" href="#E4:22_4_5_1_1_3_1_2_7_2"><span class="lab"><font color="Green" title="E111">A500</font></span></a>, <a class="ref" href="valued_0.html#D13" onmouseover="rs('valued_0/D13')" onmouseout="rh()">VALUED_0:def 13</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="E6:22_4_5_1_1_3_1_2_7_2"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c16">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c26">n</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c31">i</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c30">t</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:22_4_5_1_1_3_1_2_7_2"><span class="lab"><font color="Green" title="E109">A496</font></span></a>, <a class="txt" href="#E10:22_4_5_1_1_3"><span class="lab"><font color="Green" title="E89">A426</font></span></a>, <a class="txt" href="#E3:22_4_5_1_1"><span class="lab"><font color="Green" title="E67">A334</font></span></a>, <a class="txt" href="#E1:22_4_5_1_1"><span class="lab"><font color="Green" title="E65">A332</font></span></a>, <a class="ref" href="integra1.html#D4" onmouseover="rs('integra1/D4')" onmouseout="rh()">INTEGRA1:def 4</a></span>;<br/><a NAME="E7:22_4_5_1_1_3_1_2_7_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E113">A504</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c16">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c26">n</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c36">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c30">t</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:22_4_5_1_1_3_1_2_7_2"><span class="lab"><font color="Green" title="E112">A502</font></span></a>, <a class="ref" href="xxreal_0.html#T2" onmouseover="rs('xxreal_0/T2')" onmouseout="rh()">XXREAL_0:2</a></span>;<br/><a NAME="E8:22_4_5_1_1_3_1_2_7_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E114">A505</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c16">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c26">n</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c36">k</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c30">t</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:22_4_5_1_1_3_1_2"><span class="lab"><font color="Green" title="E98">A468</font></span></a>, <a class="ref" href="xxreal_0.html#T2" onmouseover="rs('xxreal_0/T2')" onmouseout="rh()">XXREAL_0:2</a></span>;<br/><a NAME="E9:22_4_5_1_1_3_1_2_7_2"/><span class="lab"><font color="Green" title="E115">A506</font></span>: 
( <font color="Maroon" title="c9">a</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c30">t</font> &amp; <font color="Maroon" title="c30">t</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c10">b</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_1.html#T1" onmouseover="rs('xxreal_1/T1')" onmouseout="rh()">XXREAL_1:1</a>, <a class="txt" href="#E11:22"><span class="lab"><font color="Green" title="E28">A7</font></span></a></span>;<br/><a NAME="E10:22_4_5_1_1_3_1_2_7_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E116">A507</font></span>: 
<font color="Maroon" title="c30">t</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"><a href="rcomp_1.html#K4" title="RCOMP_1:func.4">].</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c16">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c26">n</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c36">k</font></span>)</span>,<font color="Maroon" title="c10">b</font></span><a href="rcomp_1.html#K4" title="RCOMP_1:func.4">.]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:22_4_5_1_1_3_1_2_7_2"><span class="lab"><font color="Green" title="E113">A504</font></span></a></span>;<br/><span class="kw">then </span><span class="lab"><font color="Green" title="E117">A509</font></span>: <font color="Maroon" title="c46">f1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c30">t</font> = 
<span class="p1">(<span class="default"><span class="p2"><a href="rcomp_1.html#K4" title="RCOMP_1:func.4">].</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c16">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c26">n</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c36">k</font></span>)</span>,<font color="Maroon" title="c10">b</font></span><a href="rcomp_1.html#K4" title="RCOMP_1:func.4">.]</a></span> <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c30">t</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:22_4_5_1_1_3"><span class="lab"><font color="Green" title="E93">B30</font></span></a>, <a class="ref" href="funct_4.html#T13" onmouseover="rs('funct_4/T13')" onmouseout="rh()">FUNCT_4:13</a></span>
<br/>.= 
 <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:22_4_5_1_1_3_1_2_7_2"><span class="lab"><font color="Green" title="E116">A507</font></span></a>, <a class="ref" href="funcop_1.html#T7" onmouseover="rs('funcop_1/T7')" onmouseout="rh()">FUNCOP_1:7</a></span>
;<br/><a NAME="E12:22_4_5_1_1_3_1_2_7_2"/><span class="lab"><font color="Green" title="E118">A508</font></span>: 
<font color="Maroon" title="c30">t</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"><a href="rcomp_1.html#K4" title="RCOMP_1:func.4">].</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c16">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c26">n</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c36">k</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c10">b</font></span><a href="rcomp_1.html#K4" title="RCOMP_1:func.4">.]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:22_4_5_1_1_3_1_2_7_2"><span class="lab"><font color="Green" title="E114">A505</font></span></a>, <a class="txt" href="#E9:22_4_5_1_1_3_1_2_7_2"><span class="lab"><font color="Green" title="E115">A506</font></span></a></span>;<br/><span class="kw">then </span><font color="Maroon" title="c47">f2</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c30">t</font> = 
<span class="p1">(<span class="default"><span class="p2"><a href="rcomp_1.html#K4" title="RCOMP_1:func.4">].</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c16">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c26">n</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c36">k</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c10">b</font></span><a href="rcomp_1.html#K4" title="RCOMP_1:func.4">.]</a></span> <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c30">t</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:22_4_5_1_1_3"><span class="lab"><font color="Green" title="E93">B30</font></span></a>, <a class="ref" href="funct_4.html#T13" onmouseover="rs('funct_4/T13')" onmouseout="rh()">FUNCT_4:13</a></span>
<br/>.= 
 <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:22_4_5_1_1_3_1_2_7_2"><span class="lab"><font color="Green" title="E118">A508</font></span></a>, <a class="ref" href="funcop_1.html#T7" onmouseover="rs('funcop_1/T7')" onmouseout="rh()">FUNCOP_1:7</a></span>
;<br/><span class="kw">hence </span><a NAME="E14:22_4_5_1_1_3_1_2_7_2"/>
<font color="Maroon" title="c33">z</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c36">k</font> <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="txt" href="#E8:22_4_5_1_1_3"><span class="lab"><font color="Green" title="E87">A423</font></span></a>, <a class="txt" href="#E24:22_4_5_1_1_3_1_2"><span class="lab"><font color="Green" title="E108">A482</font></span></a>, <a class="txt" href="#E11:22_4_5_1_1_3_1_2_7_2"><span class="lab"><font color="Green" title="E117">A509</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></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

</div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E27:22_4_5_1_1"/>
<font color="Maroon" title="c29">g1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c30">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c27">p2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c31">i</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:22_4"><span class="lab"><font color="Green" title="E51">A317</font></span></a>, <a class="txt" href="#E5:22_4_5_1_1"><span class="lab"><font color="Green" title="E68">A336</font></span></a>, <a class="ref" href="funct_1.html#T47" onmouseover="rs('funct_1/T47')" onmouseout="rh()">FUNCT_1:47</a>, <a class="txt" href="#E13:22_4_5_1_1"><span class="lab"><font color="Green" title="E75">A348</font></span></a>, <a class="txt" href="#E6:22_4_5_1_1"><span class="lab"><font color="Green" title="E69">A337</font></span></a>, <a class="txt" href="#E12:22_4_5_1_1"><span class="lab"><font color="Green" title="E74">A347</font></span></a>, <a class="txt" href="#E25:22_4_5_1_1"><span class="lab"><font color="Green" title="E83">A357</font></span></a>, <a class="ref" href="integr23.html#T6" onmouseover="rs('integr23/T6')" onmouseout="rh()">INTEGR23:6</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="E15:22_4_5_1"/><span class="lab"><font color="Green" title="E65">A511</font></span>: 
 for <font color="Olive" title="b1">t</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c1">A</font>  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"><font color="Maroon" title="c29">g1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">t</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c21">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">t</font></span>)</span></span>)</span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c23">r</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="22_4_5_1_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c30">t</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c1">A</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"><span class="p3">(<span class="default"><font color="Maroon" title="c29">g1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c30">t</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c21">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c30">t</font></span>)</span></span>)</span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c23">r</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2</span><br/>

<a NAME="E1:22_4_5_1_2"/>
<font color="Maroon" title="c1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><span class="p2">(<span class="default"><a href="seq_4.html#K3" title="SEQ_4:func.3">lower_bound</a> <font color="Maroon" title="c1">A</font></span>)</span>,<span class="p2">(<span class="default"><a href="seq_4.html#K2" title="SEQ_4:func.2">upper_bound</a> <font color="Maroon" title="c1">A</font></span>)</span></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="integra1.html#T4" onmouseover="rs('integra1/T4')" onmouseout="rh()">INTEGRA1:4</a></span>;<br/>
<a NAME="E2:22_4_5_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E66">A512</font></span>: 
 <a href="seq_4.html#K3" title="SEQ_4:func.3">lower_bound</a> <font color="Maroon" title="c1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">a</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:22"><span class="lab"><font color="Green" title="E28">A7</font></span></a>, <a class="ref" href="integra1.html#T5" onmouseover="rs('integra1/T5')" onmouseout="rh()">INTEGRA1:5</a></span>;<br/>
<a NAME="E3:22_4_5_1_2"/>
<font color="Maroon" title="c9">a</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c26">n</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:22_4"><span class="lab"><font color="Green" title="E42">A264</font></span></a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c31">j</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a> <span class="kw"> such that </span><br/><a NAME="E5:22_4_5_1_2"/><span class="lab"><font color="Green" title="E67">A540</font></span>: 
<font color="Maroon" title="c31">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="c16">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c26">n</font></span>)</span>
 <span class="kw">and </span><br/><a NAME="E6:22_4_5_1_2"/><span class="lab"><font color="Green" title="E68">A541</font></span>: 
<font color="Maroon" title="c30">t</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c16">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c26">n</font></span>)</span>,<font color="Maroon" title="c31">j</font>)
 <span class="kw">and </span><br/><a NAME="E7:22_4_5_1_2"/><span class="lab"><font color="Green" title="E69">A542</font></span>: 
( <font color="Maroon" title="c31">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or (  <a href="seq_4.html#K3" title="SEQ_4:func.3">lower_bound</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c16">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c26">n</font></span>)</span>,<font color="Maroon" title="c31">j</font>)</span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c30">t</font> &amp; <font color="Maroon" title="c30">t</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="seq_4.html#K2" title="SEQ_4:func.2">upper_bound</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c16">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c26">n</font></span>)</span>,<font color="Maroon" title="c31">j</font>)</span>)</span> ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:22_4_5_1_2"><span class="lab"><font color="Green" title="E66">A512</font></span></a>, <a class="ref" href="dualsp05.html#T13" target="_self" onmouseover="rs('dualsp05/T13')" onmouseout="rh()">LM94</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c32">i</font> = <font color="Maroon" title="c31">j</font> as   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> ;<br/>
<span class="kw">set </span><font color="Maroon" title="c33">tD</font> = <font color="Maroon" title="c27">p2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c32">i</font>;<br/>
<a NAME="E9:22_4_5_1_2"/><span class="lab"><font color="Green" title="E70">A543</font></span>: 
 for <font color="Olive" title="b1">tD</font>, <font color="Olive" title="b2">t</font> being   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a>  st <font color="Olive" title="b1">tD</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> <font color="Maroon" title="c21">g</font> &amp; <font color="Olive" title="b2">t</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> <font color="Maroon" title="c21">g</font> &amp; <span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b1">tD</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Olive" title="b2">t</font></span>)</span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c24">s</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"><span class="p3">(<span class="default"><font color="Maroon" title="c21">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">tD</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c21">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">t</font></span>)</span></span>)</span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c23">r</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="22_4_5_1_2_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c34">tD</font>, <font color="Maroon" title="c35">t</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="c34">tD</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> <font color="Maroon" title="c21">g</font> &amp; <font color="Maroon" title="c35">t</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> <font color="Maroon" title="c21">g</font> &amp; <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="c34">tD</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c35">t</font></span>)</span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c24">s</font> 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"><font color="Maroon" title="c21">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c34">tD</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c21">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c35">t</font></span>)</span></span>)</span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c23">r</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2 )</span><br/>

<span class="kw">assume </span><a NAME="E1:22_4_5_1_2_1"/><span class="lab"><font color="Green" title="E71">A544</font></span>: 
( <font color="Maroon" title="c34">tD</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> <font color="Maroon" title="c21">g</font> &amp; <font color="Maroon" title="c35">t</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> <font color="Maroon" title="c21">g</font> &amp; <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="c34">tD</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c35">t</font></span>)</span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c24">s</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"><span class="p3">(<span class="default"><font color="Maroon" title="c21">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c34">tD</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c21">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c35">t</font></span>)</span></span>)</span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c23">r</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2</span><br/>

<a NAME="E2:22_4_5_1_2_1"/><span class="lab"><font color="Green" title="E72">B545</font></span>: 
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c21">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">u</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c1">A</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:22_4"><span class="lab"><font color="Green" title="E38">A260</font></span></a>, <a class="ref" href="relat_1.html#T62" onmouseover="rs('relat_1/T62')" onmouseout="rh()">RELAT_1:62</a>, <a class="txt" href="#E20:22_4"><span class="lab"><font color="Green" title="E51">A317</font></span></a></span>;<br/>
<a NAME="E3:22_4_5_1_2_1"/><span class="kw">then </span>
( <span class="p1">(<span class="default"><font color="Maroon" title="c15">u</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c34">tD</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">u</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c34">tD</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c15">u</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c35">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">u</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c35">t</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:22_4_5_1_2_1"><span class="lab"><font color="Green" title="E71">A544</font></span></a>, <a class="ref" href="funct_1.html#T47" onmouseover="rs('funct_1/T47')" onmouseout="rh()">FUNCT_1:47</a></span>;<br/>
<span class="kw">hence </span><a NAME="E4:22_4_5_1_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"><font color="Maroon" title="c21">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c34">tD</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c21">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c35">t</font></span>)</span></span>)</span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c23">r</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:22_4"><span class="lab"><font color="Green" title="E51">A317</font></span></a>, <a class="txt" href="#E2:22_4_5_1_2_1"><span class="lab"><font color="Green" title="E72">B545</font></span></a>, <a class="txt" href="#E5:22_4_5"><span class="lab"><font color="Green" title="E56">A323</font></span></a>, <a class="txt" href="#E1:22_4_5_1_2_1"><span class="lab"><font color="Green" title="E71">A544</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="E10:22_4_5_1_2"/>
<font color="Maroon" title="c27">p2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c32">i</font> <a href="tarski.html#R2" title="TARSKI: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="c15">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="c16">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c26">n</font></span>)</span>,<font color="Maroon" title="c32">i</font>)</span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:22_4_5_1"><span class="lab"><font color="Green" title="E60">A327</font></span></a>, <a class="txt" href="#E5:22_4_5_1_2"><span class="lab"><font color="Green" title="E67">A540</font></span></a></span>;<br/>
<a NAME="E11:22_4_5_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c27">p2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c32">i</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c15">u</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c16">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c26">n</font></span>)</span>,<font color="Maroon" title="c32">i</font>)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#T61" onmouseover="rs('relat_1/T61')" onmouseout="rh()">RELAT_1:61</a></span>;<br/>
<a NAME="E12:22_4_5_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E71">A549</font></span>: 
( <font color="Maroon" title="c27">p2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c32">i</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c15">u</font> &amp; <font color="Maroon" title="c27">p2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c32">i</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c16">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c26">n</font></span>)</span>,<font color="Maroon" title="c32">i</font>) )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D4" onmouseover="rs('xboole_0/D4')" onmouseout="rh()">XBOOLE_0:def 4</a></span>;<br/>
<a NAME="E13:22_4_5_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E72">B551</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"><font color="Maroon" title="c27">p2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c32">i</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c30">t</font></span>)</span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c24">s</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:22_4_5_1"><span class="lab"><font color="Green" title="E62">A329</font></span></a>, <a class="txt" href="#E5:22_4_5_1_2"><span class="lab"><font color="Green" title="E67">A540</font></span></a>, <a class="txt" href="#E6:22_4_5_1_2"><span class="lab"><font color="Green" title="E68">A541</font></span></a>, <a class="ref" href="integr20.html#T12" onmouseover="rs('integr20/T12')" onmouseout="rh()">INTEGR20:12</a></span>;<br/>
<a NAME="E14:22_4_5_1_2"/>
<font color="Maroon" title="c29">g1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c30">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c27">p2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c32">i</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:22_4_5_1"><span class="lab"><font color="Green" title="E64">A331</font></span></a>, <a class="txt" href="#E5:22_4_5_1_2"><span class="lab"><font color="Green" title="E67">A540</font></span></a>, <a class="txt" href="#E6:22_4_5_1_2"><span class="lab"><font color="Green" title="E68">A541</font></span></a>, <a class="txt" href="#E7:22_4_5_1_2"><span class="lab"><font color="Green" title="E69">A542</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E15:22_4_5_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"><font color="Maroon" title="c29">g1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c30">t</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c21">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c30">t</font></span>)</span></span>)</span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c23">r</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:22_4_5_1_2"><span class="lab"><font color="Green" title="E72">B551</font></span></a>, <a class="txt" href="#E9:22_4_5_1_2"><span class="lab"><font color="Green" title="E70">A543</font></span></a>, <a class="txt" href="#E12:22_4_5_1_2"><span class="lab"><font color="Green" title="E71">A549</font></span></a>, <a class="txt" href="#E1:22_4"><span class="lab"><font color="Green" title="E38">A260</font></span></a>, <a class="txt" href="#E20:22_4"><span class="lab"><font color="Green" title="E51">A317</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="c30">z</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c19">xD</font> <a href="normsp_0.html#K1" title="NORMSP_0:func.1">.</a> <font color="Maroon" title="c26">n</font></span>)</span> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <font color="Maroon" title="c22">v0</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="c0sp1.html#K11" title="C0SP1:func.11">R_Normed_Algebra_of_BoundedFunctions</a> <font color="Maroon" title="c1">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="c31">g0</font> = <font color="Maroon" title="c30">z</font> as   <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> <span class="kw">by</span> <span class="lab"><a class="ref" href="dualsp05.html#T11" target="_self" onmouseover="rs('dualsp05/T11')" onmouseout="rh()">LM88</a></span>;<br/>
<a NAME="E18:22_4_5_1"/>
<font color="Maroon" title="c31">g0</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="c0sp1.html#K6" title="C0SP1:func.6">BoundedFunctions</a> <font color="Maroon" title="c1">A</font>
 
;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c32">g2</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><span class="kw"> such that </span><br/><a NAME="E20:22_4_5_1"/><span class="lab"><font color="Green" title="E66">A552</font></span>: 
( <font color="Maroon" title="c32">g2</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c31">g0</font> &amp; <font color="Maroon" title="c32">g2</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c1">A</font> is  <a href="comseq_2.html#V1" title="COMSEQ_2:attr.1">bounded</a>  )
 ;<br/>
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="22_4_5_1_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="c0sp1.html#K9" title="C0SP1:func.9">PreNorms</a> <font color="Maroon" title="c31">g0</font> holds <br/><font color="Olive" title="b1">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c23">r</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2</span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c33">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="c33">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="c0sp1.html#K9" title="C0SP1:func.9">PreNorms</a> <font color="Maroon" title="c31">g0</font> implies <font color="Maroon" title="c33">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c23">r</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2 )</span><br/><span class="kw">assume </span><a NAME="E1:22_4_5_1_3"/>
<font color="Maroon" title="c33">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="c0sp1.html#K9" title="C0SP1:func.9">PreNorms</a> <font color="Maroon" title="c31">g0</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="c33">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c23">r</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2</span><br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c34">t</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c1">A</font><span class="kw"> such that </span><br/><a NAME="E3:22_4_5_1_3"/><span class="lab"><font color="Green" title="E67">A553</font></span>: 
<font color="Maroon" title="c33">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="c31">g0</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c34">t</font></span>)</span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span>
 ;<br/><a NAME="E4:22_4_5_1_3"/>
<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"><font color="Maroon" title="c29">g1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c34">t</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c21">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c34">t</font></span>)</span></span>)</span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c23">r</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:22_4_5_1"><span class="lab"><font color="Green" title="E65">A511</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E5:22_4_5_1_3"/>
<font color="Maroon" title="c33">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c23">r</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:22_4_5_1_3"><span class="lab"><font color="Green" title="E67">A553</font></span></a>, <a class="txt" href="#E2:22"><span class="lab"><font color="Green" title="E23">A2</font></span></a>, <a class="txt" href="#E20:22_4"><span class="lab"><font color="Green" title="E51">A317</font></span></a>, <a class="txt" href="#E13:22_4_5_1"><span class="lab"><font color="Green" title="E63">A330</font></span></a>, <a class="ref" href="c0sp1.html#T34" onmouseover="rs('c0sp1/T34')" onmouseout="rh()">C0SP1:34</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="E22:22_4_5_1"/><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="c0sp1.html#K9" title="C0SP1:func.9">PreNorms</a> <font color="Maroon" title="c31">g0</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c23">r</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2
 
<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/>
<a NAME="E23:22_4_5_1"/><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="c30">z</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> <font color="Maroon" title="c23">r</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:22_4_5_1"><span class="lab"><font color="Green" title="E66">A552</font></span></a>, <a class="ref" href="c0sp1.html#T20" onmouseover="rs('c0sp1/T20')" onmouseout="rh()">C0SP1:20</a></span>;<br/>
<a NAME="E24:22_4_5_1"/><span class="kw">then </span>
<span class="p1"><a href="normsp_0.html#K2" title="NORMSP_0:func.2">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c19">xD</font> <a href="normsp_0.html#K1" title="NORMSP_0:func.1">.</a> <font color="Maroon" title="c26">n</font></span>)</span> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <font color="Maroon" title="c22">v0</font></span>)</span></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> <font color="Maroon" title="c23">r</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2
 
<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">hence </span><a NAME="E25:22_4_5_1"/>
<span class="p1"><a href="normsp_0.html#K2" title="NORMSP_0:func.2">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c19">xD</font> <a href="normsp_0.html#K1" title="NORMSP_0:func.1">.</a> <font color="Maroon" title="c26">n</font></span>)</span> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <font color="Maroon" title="c22">v0</font></span>)</span></span><a href="normsp_0.html#K2" title="NORMSP_0:func.2">.||</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c23">r</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:22_4_5"><span class="lab"><font color="Green" title="E54">A321</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>
<span class="kw">take </span>
<font color="Maroon" title="c25">m0</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">  for <font color="Olive" title="b1">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Maroon" title="c25">m0</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">n</font> holds <br/><span class="p1"><a href="normsp_0.html#K2" title="NORMSP_0:func.2">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c19">xD</font> <a href="normsp_0.html#K1" title="NORMSP_0:func.1">.</a> <font color="Olive" title="b1">n</font></span>)</span> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <font color="Maroon" title="c22">v0</font></span>)</span></span><a href="normsp_0.html#K2" title="NORMSP_0:func.2">.||</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c23">r</font></span><br/>

<span class="kw">thus </span><a NAME="E9:22_4_5"/>
 for <font color="Olive" title="b1">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Maroon" title="c25">m0</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">n</font> holds <br/><span class="p1"><a href="normsp_0.html#K2" title="NORMSP_0:func.2">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c19">xD</font> <a href="normsp_0.html#K1" title="NORMSP_0:func.1">.</a> <font color="Olive" title="b1">n</font></span>)</span> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <font color="Maroon" title="c22">v0</font></span>)</span></span><a href="normsp_0.html#K2" title="NORMSP_0:func.2">.||</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c23">r</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:22_4_5"><span class="lab"><font color="Green" title="E58">A325</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="E23:22_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E53">A555</font></span>: 
<font color="Maroon" title="c19">xD</font> is  <a href="normsp_1.html#V3" title="NORMSP_1:attr.3">convergent</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="normsp_1.html#D6" onmouseover="rs('normsp_1/D6')" onmouseout="rh()">NORMSP_1:def 6</a></span>;<br/>
<a NAME="E24:22_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E54">A556</font></span>: 
 <a href="normsp_1.html#K5" title="NORMSP_1:func.5">lim</a> <font color="Maroon" title="c19">xD</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">v</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E22:22_4"><span class="lab"><font color="Green" title="E52">A318</font></span></a>, <a class="ref" href="normsp_1.html#D7" onmouseover="rs('normsp_1/D7')" onmouseout="rh()">NORMSP_1:def 7</a></span>;<br/>
<a NAME="E25:22_4"/>
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c7">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="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="funct_2.html#D1" onmouseover="rs('funct_2/D1')" onmouseout="rh()">FUNCT_2:def 1</a></span>;<br/>
<a NAME="E26:22_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E55">A557</font></span>: 
 <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c19">xD</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c7">f</font>
 
;<br/>
<a NAME="E27:22_4"/>
<font color="Maroon" title="c20">v</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="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>
 
;<br/>
<a NAME="E28:22_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E56">A558</font></span>: 
<font color="Maroon" title="c20">v</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c7">f</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D1" onmouseover="rs('funct_2/D1')" onmouseout="rh()">FUNCT_2:def 1</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c23">K</font> being   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a><span class="kw"> such that </span><br/><a NAME="E30:22_4"/><span class="lab"><font color="Green" title="E57">A559</font></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> <font color="Maroon" title="c23">K</font> &amp; (  for <font color="Olive" title="b1">x</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="c0sp1.html#K11" title="C0SP1:func.11">R_Normed_Algebra_of_BoundedFunctions</a>  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> holds  <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="c7">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">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> <font color="Maroon" title="c23">K</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1"><a href="normsp_0.html#K2" title="NORMSP_0:func.2">||.</a><span class="default"><font color="Olive" title="b1">x</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="ref" href="dualsp01.html#D9" onmouseover="rs('dualsp01/D9')" onmouseout="rh()">DUALSP01:def 9</a></span>;<br/>
<a NAME="E31:22_4"/>
 for <font color="Olive" title="b1">r</font> being   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a>  st  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">r</font> holds <br/> ex <font color="Olive" title="b2">s</font> being   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a> st <br/>(  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b2">s</font> &amp; (  for <font color="Olive" title="b3">x1</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="c0sp1.html#K11" title="C0SP1:func.11">R_Normed_Algebra_of_BoundedFunctions</a>  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>  st <font color="Olive" title="b3">x1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c7">f</font> &amp; <span class="p1"><a href="normsp_0.html#K2" title="NORMSP_0:func.2">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b3">x1</font> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <font color="Maroon" title="c20">v</font></span>)</span></span><a href="normsp_0.html#K2" title="NORMSP_0:func.2">.||</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b2">s</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"><span class="p3">(<span class="default"><font color="Maroon" title="c7">f</font> <a href="funct_2.html#K10" title="FUNCT_2:func.10">/.</a> <font color="Olive" title="b3">x1</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">f</font> <a href="funct_2.html#K10" title="FUNCT_2:func.10">/.</a> <font color="Maroon" title="c20">v</font></span>)</span></span>)</span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">r</font> ) )
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="22_4_6"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c24">r</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"> (  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c24">r</font> implies  ex <font color="Olive" title="b1">s</font> being   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a> st <br/>(  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">s</font> &amp; (  for <font color="Olive" title="b2">x1</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="c0sp1.html#K11" title="C0SP1:func.11">R_Normed_Algebra_of_BoundedFunctions</a>  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>  st <font color="Olive" title="b2">x1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c7">f</font> &amp; <span class="p1"><a href="normsp_0.html#K2" title="NORMSP_0:func.2">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b2">x1</font> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <font color="Maroon" title="c20">v</font></span>)</span></span><a href="normsp_0.html#K2" title="NORMSP_0:func.2">.||</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">s</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"><span class="p3">(<span class="default"><font color="Maroon" title="c7">f</font> <a href="funct_2.html#K10" title="FUNCT_2:func.10">/.</a> <font color="Olive" title="b2">x1</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">f</font> <a href="funct_2.html#K10" title="FUNCT_2:func.10">/.</a> <font color="Maroon" title="c20">v</font></span>)</span></span>)</span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c24">r</font> ) ) )</span><br/>

<span class="kw">assume </span><a NAME="E1:22_4_6"/><span class="lab"><font color="Green" title="E58">A561</font></span>: 
 <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c24">r</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">  ex <font color="Olive" title="b1">s</font> being   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a> st <br/>(  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">s</font> &amp; (  for <font color="Olive" title="b2">x1</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="c0sp1.html#K11" title="C0SP1:func.11">R_Normed_Algebra_of_BoundedFunctions</a>  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>  st <font color="Olive" title="b2">x1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c7">f</font> &amp; <span class="p1"><a href="normsp_0.html#K2" title="NORMSP_0:func.2">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b2">x1</font> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <font color="Maroon" title="c20">v</font></span>)</span></span><a href="normsp_0.html#K2" title="NORMSP_0:func.2">.||</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">s</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"><span class="p3">(<span class="default"><font color="Maroon" title="c7">f</font> <a href="funct_2.html#K10" title="FUNCT_2:func.10">/.</a> <font color="Olive" title="b2">x1</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">f</font> <a href="funct_2.html#K10" title="FUNCT_2:func.10">/.</a> <font color="Maroon" title="c20">v</font></span>)</span></span>)</span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c24">r</font> ) )</span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c25">s</font> = <font color="Maroon" title="c24">r</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c23">K</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span> as   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a> ;<br/>
<a NAME="E3:22_4_6"/><span class="lab"><font color="Green" title="E59">A563</font></span>: 
 for <font color="Olive" title="b1">x1</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="c0sp1.html#K11" title="C0SP1:func.11">R_Normed_Algebra_of_BoundedFunctions</a>  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>  st <font color="Olive" title="b1">x1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c7">f</font> &amp; <span class="p1"><a href="normsp_0.html#K2" title="NORMSP_0:func.2">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b1">x1</font> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <font color="Maroon" title="c20">v</font></span>)</span></span><a href="normsp_0.html#K2" title="NORMSP_0:func.2">.||</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c25">s</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"><span class="p3">(<span class="default"><font color="Maroon" title="c7">f</font> <a href="funct_2.html#K10" title="FUNCT_2:func.10">/.</a> <font color="Olive" title="b1">x1</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">f</font> <a href="funct_2.html#K10" title="FUNCT_2:func.10">/.</a> <font color="Maroon" title="c20">v</font></span>)</span></span>)</span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c24">r</font>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="22_4_6_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c26">x1</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="c0sp1.html#K11" title="C0SP1:func.11">R_Normed_Algebra_of_BoundedFunctions</a>  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>; <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="c26">x1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c7">f</font> &amp; <span class="p1"><a href="normsp_0.html#K2" title="NORMSP_0:func.2">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c26">x1</font> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <font color="Maroon" title="c20">v</font></span>)</span></span><a href="normsp_0.html#K2" title="NORMSP_0:func.2">.||</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c25">s</font> 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"><font color="Maroon" title="c7">f</font> <a href="funct_2.html#K10" title="FUNCT_2:func.10">/.</a> <font color="Maroon" title="c26">x1</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">f</font> <a href="funct_2.html#K10" title="FUNCT_2:func.10">/.</a> <font color="Maroon" title="c20">v</font></span>)</span></span>)</span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c24">r</font> )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:22_4_6_1"/>
<font color="Maroon" title="c26">x1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c7">f</font>
 <span class="kw">and </span><br/><a NAME="E2:22_4_6_1"/><span class="lab"><font color="Green" title="E60">A565</font></span>: 
<span class="p1"><a href="normsp_0.html#K2" title="NORMSP_0:func.2">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c26">x1</font> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <font color="Maroon" title="c20">v</font></span>)</span></span><a href="normsp_0.html#K2" title="NORMSP_0:func.2">.||</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c25">s</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"><span class="p3">(<span class="default"><font color="Maroon" title="c7">f</font> <a href="funct_2.html#K10" title="FUNCT_2:func.10">/.</a> <font color="Maroon" title="c26">x1</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">f</font> <a href="funct_2.html#K10" title="FUNCT_2:func.10">/.</a> <font color="Maroon" title="c20">v</font></span>)</span></span>)</span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c24">r</font></span><br/>

<a NAME="E3:22_4_6_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"><font color="Maroon" title="c7">f</font> <a href="funct_2.html#K10" title="FUNCT_2:func.10">/.</a> <font color="Maroon" title="c26">x1</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">f</font> <a href="funct_2.html#K10" title="FUNCT_2:func.10">/.</a> <font color="Maroon" title="c20">v</font></span>)</span></span>)</span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span> <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="c7">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c26">x1</font> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <font color="Maroon" title="c20">v</font></span>)</span></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="hahnban.html#T19" onmouseover="rs('hahnban/T19')" onmouseout="rh()">HAHNBAN:19</a></span>;<br/>
<a NAME="E4:22_4_6_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E61">A567</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"><font color="Maroon" title="c7">f</font> <a href="funct_2.html#K10" title="FUNCT_2:func.10">/.</a> <font color="Maroon" title="c26">x1</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">f</font> <a href="funct_2.html#K10" title="FUNCT_2:func.10">/.</a> <font color="Maroon" title="c20">v</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> <font color="Maroon" title="c23">K</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1"><a href="normsp_0.html#K2" title="NORMSP_0:func.2">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c26">x1</font> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <font color="Maroon" title="c20">v</font></span>)</span></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="#E30:22_4"><span class="lab"><font color="Green" title="E57">A559</font></span></a></span>;<br/>
<a NAME="E5:22_4_6_1"/>
<font color="Maroon" title="c23">K</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c23">K</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T145" onmouseover="rs('xreal_1/T145')" onmouseout="rh()">XREAL_1:145</a></span>;<br/>
<a NAME="E6:22_4_6_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E62">A569</font></span>: 
<font color="Maroon" title="c23">K</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1"><a href="normsp_0.html#K2" title="NORMSP_0:func.2">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c26">x1</font> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <font color="Maroon" title="c20">v</font></span>)</span></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> <span class="p1">(<span class="default"><font color="Maroon" title="c23">K</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1"><a href="normsp_0.html#K2" title="NORMSP_0:func.2">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c26">x1</font> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <font color="Maroon" title="c20">v</font></span>)</span></span><a href="normsp_0.html#K2" title="NORMSP_0:func.2">.||</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T64" onmouseover="rs('xreal_1/T64')" onmouseout="rh()">XREAL_1:64</a></span>;<br/>
<a NAME="E7:22_4_6_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c23">K</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1"><a href="normsp_0.html#K2" title="NORMSP_0:func.2">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c26">x1</font> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <font color="Maroon" title="c20">v</font></span>)</span></span><a href="normsp_0.html#K2" title="NORMSP_0:func.2">.||</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c23">K</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c25">s</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E30:22_4"><span class="lab"><font color="Green" title="E57">A559</font></span></a>, <a class="txt" href="#E2:22_4_6_1"><span class="lab"><font color="Green" title="E60">A565</font></span></a>, <a class="ref" href="xreal_1.html#T68" onmouseover="rs('xreal_1/T68')" onmouseout="rh()">XREAL_1:68</a></span>;<br/>
<a NAME="E8:22_4_6_1"/><span class="kw">then </span>
<font color="Maroon" title="c23">K</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1"><a href="normsp_0.html#K2" title="NORMSP_0:func.2">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c26">x1</font> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <font color="Maroon" title="c20">v</font></span>)</span></span><a href="normsp_0.html#K2" title="NORMSP_0:func.2">.||</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c23">K</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c25">s</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:22_4_6_1"><span class="lab"><font color="Green" title="E62">A569</font></span></a>, <a class="ref" href="xxreal_0.html#T2" onmouseover="rs('xxreal_0/T2')" onmouseout="rh()">XXREAL_0:2</a></span>;<br/>
<a NAME="E9:22_4_6_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"><span class="p3">(<span class="default"><font color="Maroon" title="c7">f</font> <a href="funct_2.html#K10" title="FUNCT_2:func.10">/.</a> <font color="Maroon" title="c26">x1</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">f</font> <a href="funct_2.html#K10" title="FUNCT_2:func.10">/.</a> <font color="Maroon" title="c20">v</font></span>)</span></span>)</span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c23">K</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c25">s</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:22_4_6_1"><span class="lab"><font color="Green" title="E61">A567</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="E10:22_4_6_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"><font color="Maroon" title="c7">f</font> <a href="funct_2.html#K10" title="FUNCT_2:func.10">/.</a> <font color="Maroon" title="c26">x1</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">f</font> <a href="funct_2.html#K10" title="FUNCT_2:func.10">/.</a> <font color="Maroon" title="c20">v</font></span>)</span></span>)</span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c24">r</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E30:22_4"><span class="lab"><font color="Green" title="E57">A559</font></span></a>, <a class="ref" href="xcmplx_1.html#T87" onmouseover="rs('xcmplx_1/T87')" onmouseout="rh()">XCMPLX_1:87</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">take </span>
<font color="Maroon" title="c25">s</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"> (  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c25">s</font> &amp; (  for <font color="Olive" title="b1">x1</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="c0sp1.html#K11" title="C0SP1:func.11">R_Normed_Algebra_of_BoundedFunctions</a>  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>  st <font color="Olive" title="b1">x1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c7">f</font> &amp; <span class="p1"><a href="normsp_0.html#K2" title="NORMSP_0:func.2">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b1">x1</font> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <font color="Maroon" title="c20">v</font></span>)</span></span><a href="normsp_0.html#K2" title="NORMSP_0:func.2">.||</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c25">s</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"><span class="p3">(<span class="default"><font color="Maroon" title="c7">f</font> <a href="funct_2.html#K10" title="FUNCT_2:func.10">/.</a> <font color="Olive" title="b1">x1</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">f</font> <a href="funct_2.html#K10" title="FUNCT_2:func.10">/.</a> <font color="Maroon" title="c20">v</font></span>)</span></span>)</span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c24">r</font> ) )</span><br/>

<span class="kw">thus </span><a NAME="E4:22_4_6"/>
(  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c25">s</font> &amp; (  for <font color="Olive" title="b1">x1</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="c0sp1.html#K11" title="C0SP1:func.11">R_Normed_Algebra_of_BoundedFunctions</a>  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>  st <font color="Olive" title="b1">x1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c7">f</font> &amp; <span class="p1"><a href="normsp_0.html#K2" title="NORMSP_0:func.2">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b1">x1</font> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <font color="Maroon" title="c20">v</font></span>)</span></span><a href="normsp_0.html#K2" title="NORMSP_0:func.2">.||</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c25">s</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"><span class="p3">(<span class="default"><font color="Maroon" title="c7">f</font> <a href="funct_2.html#K10" title="FUNCT_2:func.10">/.</a> <font color="Olive" title="b1">x1</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">f</font> <a href="funct_2.html#K10" title="FUNCT_2:func.10">/.</a> <font color="Maroon" title="c20">v</font></span>)</span></span>)</span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c24">r</font> ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E30:22_4"><span class="lab"><font color="Green" title="E57">A559</font></span></a>, <a class="txt" href="#E1:22_4_6"><span class="lab"><font color="Green" title="E58">A561</font></span></a>, <a class="ref" href="xreal_1.html#T139" onmouseover="rs('xreal_1/T139')" onmouseout="rh()">XREAL_1:139</a>, <a class="txt" href="#E3:22_4_6"><span class="lab"><font color="Green" title="E59">A563</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="E32:22_4"/><span class="kw">then </span>
<font color="Maroon" title="c7">f</font> <a href="nfcont_1.html#R2" title="NFCONT_1:pred.2">is_continuous_in</a> <font color="Maroon" title="c20">v</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E28:22_4"><span class="lab"><font color="Green" title="E56">A558</font></span></a>, <a class="ref" href="nfcont_1.html#T8" onmouseover="rs('nfcont_1/T8')" onmouseout="rh()">NFCONT_1:8</a></span>;<br/>
<a NAME="E33:22_4"/><span class="kw">then </span>
<font color="Maroon" title="c7">f</font> <a href="funct_2.html#K10" title="FUNCT_2:func.10">/.</a> <font color="Maroon" title="c20">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="seq_2.html#K1" title="SEQ_2:func.1">lim</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">f</font> <a href="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <font color="Maroon" title="c19">xD</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E23:22_4"><span class="lab"><font color="Green" title="E53">A555</font></span></a>, <a class="txt" href="#E24:22_4"><span class="lab"><font color="Green" title="E54">A556</font></span></a>, <a class="txt" href="#E26:22_4"><span class="lab"><font color="Green" title="E55">A557</font></span></a>, <a class="ref" href="nfcont_1.html#D6" onmouseover="rs('nfcont_1/D6')" onmouseout="rh()">NFCONT_1:def 6</a></span>;<br/>
<a NAME="E34:22_4"/><span class="kw">then </span>
 <a href="seq_2.html#K1" title="SEQ_2:func.1">lim</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">f</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c19">xD</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c20">v</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E26:22_4"><span class="lab"><font color="Green" title="E55">A557</font></span></a>, <a class="ref" href="funct_2.html#D11" onmouseover="rs('funct_2/D11')" onmouseout="rh()">FUNCT_2:def 11</a></span>;<br/>
<a NAME="E35:22_4"/><span class="kw">then </span>
<font color="Maroon" title="c7">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">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="c15">u</font>,<font color="Maroon" title="c12">rho</font>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:22_4"><span class="lab"><font color="Green" title="E48">B314</font></span></a>, <a class="ref" href="funct_2.html#D8" onmouseover="rs('funct_2/D8')" onmouseout="rh()">FUNCT_2:def 8</a>, <a class="txt" href="#E15:22_4"><span class="lab"><font color="Green" title="E49">A315</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E36:22_4"/>
<font color="Maroon" title="c2">x</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">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="c15">u</font>,<font color="Maroon" title="c12">rho</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:22"><span class="lab"><font color="Green" title="E27">A6</font></span></a>, <a class="ref" href="funct_1.html#T49" onmouseover="rs('funct_1/T49')" onmouseout="rh()">FUNCT_1:49</a>, <a class="txt" href="#E7:22_4"><span class="lab"><font color="Green" title="E43">A269</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="E30:22"/><span class="kw">then </span><span class="lab"><font color="Green" title="E38">B573</font></span>: 
<span class="p1"><a href="normsp_0.html#K2" title="NORMSP_0:func.2">||.</a><span class="default"><font color="Maroon" title="c2">x</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="c12">rho</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E22:22"><span class="lab"><font color="Green" title="E33">A251</font></span></a>, <a class="ref" href="dualsp05.html#T15" target="_self" onmouseover="rs('dualsp05/T15')" onmouseout="rh()">Lm83</a></span>;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c12">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"> ( <font color="Maroon" title="c12">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="c2">x</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="c12">rho</font>) ) &amp; <span class="p1"><a href="normsp_0.html#K2" title="NORMSP_0:func.2">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="normsp_0.html#K2" title="NORMSP_0:func.2">.||</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="integr22.html#K2" title="INTEGR22:func.2">total_vd</a> <font color="Maroon" title="c12">rho</font> )</span><br/>

<span class="kw">thus </span><a NAME="E31:22"/>
( <font color="Maroon" title="c12">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="c2">x</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="c12">rho</font>) ) &amp; <span class="p1"><a href="normsp_0.html#K2" title="NORMSP_0:func.2">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="normsp_0.html#K2" title="NORMSP_0:func.2">.||</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="integr22.html#K2" title="INTEGR22:func.2">total_vd</a> <font color="Maroon" title="c12">rho</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E21:22"><span class="lab"><font color="Green" title="E32">B250</font></span></a>, <a class="txt" href="#E29:22"><span class="lab"><font color="Green" title="E37">A259</font></span></a>, <a class="txt" href="#E30:22"><span class="lab"><font color="Green" title="E38">B573</font></span></a>, <a class="txt" href="#E28:22"><span class="lab"><font color="Green" title="E36">A258</font></span></a>, <a class="ref" href="xxreal_0.html#T1" onmouseover="rs('xxreal_0/T1')" onmouseout="rh()">XXREAL_0:1</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>
