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<span class="kw">let </span><font color="Maroon" title="c1">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">  for <font color="Olive" title="b1">f1</font>, <font color="Olive" title="b2">f2</font> being   <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>   <a href="valued_0.html#V6" title="VALUED_0:attr.6">natural-valued</a>  <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a>  st <font color="Maroon" title="c1">n</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1 &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="flexary1.html#K4" title="FLEXARY1:func.4">|^</a> <font color="Olive" title="b1">f1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1</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"><font color="Maroon" title="c1">n</font> <a href="flexary1.html#K4" title="FLEXARY1:func.4">|^</a> <font color="Olive" title="b1">f1</font></span>)</span>,2) <a href="flexary1.html#K3" title="FLEXARY1:func.3">+...</a></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="flexary1.html#K4" title="FLEXARY1:func.4">|^</a> <font color="Olive" title="b2">f2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1</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"><font color="Maroon" title="c1">n</font> <a href="flexary1.html#K4" title="FLEXARY1:func.4">|^</a> <font color="Olive" title="b2">f2</font></span>)</span>,2) <a href="flexary1.html#K3" title="FLEXARY1:func.3">+...</a></span>)</span> holds <br/> <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Olive" title="b1">f1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Olive" title="b2">f2</font></span><br/><span class="kw">let </span><font color="Maroon" title="c2">f1</font>, <font color="Maroon" title="c3">f2</font> be   <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>   <a href="valued_0.html#V6" title="VALUED_0:attr.6">natural-valued</a>  <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</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="c1">n</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1 &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="flexary1.html#K4" title="FLEXARY1:func.4">|^</a> <font color="Maroon" title="c2">f1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1</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"><font color="Maroon" title="c1">n</font> <a href="flexary1.html#K4" title="FLEXARY1:func.4">|^</a> <font color="Maroon" title="c2">f1</font></span>)</span>,2) <a href="flexary1.html#K3" title="FLEXARY1:func.3">+...</a></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="flexary1.html#K4" title="FLEXARY1:func.4">|^</a> <font color="Maroon" title="c3">f2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1</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"><font color="Maroon" title="c1">n</font> <a href="flexary1.html#K4" title="FLEXARY1:func.4">|^</a> <font color="Maroon" title="c3">f2</font></span>)</span>,2) <a href="flexary1.html#K3" title="FLEXARY1:func.3">+...</a></span>)</span> implies  <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c2">f1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c3">f2</font> )</span><br/>



<span class="kw">assume </span><a NAME="E1:45"/><span class="lab"><font color="Green" title="E31">A1</font></span>: 
( <font color="Maroon" title="c1">n</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1 &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="flexary1.html#K4" title="FLEXARY1:func.4">|^</a> <font color="Maroon" title="c2">f1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1</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"><font color="Maroon" title="c1">n</font> <a href="flexary1.html#K4" title="FLEXARY1:func.4">|^</a> <font color="Maroon" title="c2">f1</font></span>)</span>,2) <a href="flexary1.html#K3" title="FLEXARY1:func.3">+...</a></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="flexary1.html#K4" title="FLEXARY1:func.4">|^</a> <font color="Maroon" title="c3">f2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1</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"><font color="Maroon" title="c1">n</font> <a href="flexary1.html#K4" title="FLEXARY1:func.4">|^</a> <font color="Maroon" title="c3">f2</font></span>)</span>,2) <a href="flexary1.html#K3" title="FLEXARY1:func.3">+...</a></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="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c2">f1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c3">f2</font></span><br/>

<a NAME="E2:45"/><span class="lab"><font color="Green" title="E32">A2</font></span>: 
(  <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c2">f1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  &amp;  <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c3">f2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="valued_0.html#D6" onmouseover="rs('valued_0/D6')" onmouseout="rh()">VALUED_0:def 6</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c4">F1</font> = <font color="Maroon" title="c2">f1</font>, <font color="Maroon" title="c5">F2</font> = <font color="Maroon" title="c3">f2</font> as    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D4" onmouseover="rs('finseq_1/D4')" onmouseout="rh()">FINSEQ_1:def 4</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c6">s1</font> =  <a href="rfinseq2.html#K6" title="RFINSEQ2:func.6">sort_a</a> <font color="Maroon" title="c4">F1</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c7">s2</font> =  <a href="rfinseq2.html#K6" title="RFINSEQ2:func.6">sort_a</a> <font color="Maroon" title="c5">F2</font>;<br/>
<a NAME="E4:45"/><span class="lab"><font color="Green" title="E33">A3</font></span>: 
( <font color="Maroon" title="c4">F1</font>, <a href="rfinseq2.html#K6" title="RFINSEQ2:func.6">sort_a</a> <font color="Maroon" title="c4">F1</font> <a href="classes1.html#R2" title="CLASSES1:pred.2">are_fiberwise_equipotent</a>  &amp; <font color="Maroon" title="c5">F2</font>, <a href="rfinseq2.html#K6" title="RFINSEQ2:func.6">sort_a</a> <font color="Maroon" title="c5">F2</font> <a href="classes1.html#R2" title="CLASSES1:pred.2">are_fiberwise_equipotent</a>  )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="rfinseq2.html#D6" onmouseover="rs('rfinseq2/D6')" onmouseout="rh()">RFINSEQ2:def 6</a></span>;<br/>
<a NAME="E5:45"/><span class="lab"><font color="Green" title="E34">A4</font></span>: 
 <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <span class="p1">(<span class="default"><a href="rfinseq2.html#K6" title="RFINSEQ2:func.6">sort_a</a> <font color="Maroon" title="c4">F1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c2">f1</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="rfinseq2.html#D6" onmouseover="rs('rfinseq2/D6')" onmouseout="rh()">RFINSEQ2:def 6</a>, <a class="ref" href="classes1.html#T75" onmouseover="rs('classes1/T75')" onmouseout="rh()">CLASSES1:75</a></span>;<br/>
<a NAME="E6:45"/><span class="kw">then </span><span class="lab"><font color="Green" title="E35">A5</font></span>: 
 <a href="rfinseq2.html#K6" title="RFINSEQ2:func.6">sort_a</a> <font color="Maroon" title="c4">F1</font> is  <a href="valued_0.html#V6" title="VALUED_0:attr.6">natural-valued</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:45"><span class="lab"><font color="Green" title="E32">A2</font></span></a>, <a class="ref" href="valued_0.html#D6" onmouseover="rs('valued_0/D6')" onmouseout="rh()">VALUED_0:def 6</a></span>;<br/>
<a NAME="E7:45"/>
<font color="Maroon" title="c1">n</font> <a href="flexary1.html#K4" title="FLEXARY1:func.4">|^</a> <font color="Maroon" title="c4">F1</font>,<font color="Maroon" title="c1">n</font> <a href="flexary1.html#K4" title="FLEXARY1:func.4">|^</a> <span class="p1">(<span class="default"><a href="rfinseq2.html#K6" title="RFINSEQ2:func.6">sort_a</a> <font color="Maroon" title="c4">F1</font></span>)</span> <a href="classes1.html#R2" title="CLASSES1:pred.2">are_fiberwise_equipotent</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:45"><span class="lab"><font color="Green" title="E33">A3</font></span></a>, <a class="txt" href="#E1:45"><span class="lab"><font color="Green" title="E31">A1</font></span></a>, <a class="txt" href="#E6:45"><span class="lab"><font color="Green" title="E35">A5</font></span></a>, <a class="ref" href="flexary1.html#T29" target="_self" onmouseover="rs('flexary1/T29')" onmouseout="rh()">Th29</a></span>;<br/>
<a NAME="E8:45"/><span class="kw">then </span><span class="lab"><font color="Green" title="E36">A8</font></span>: 
 <a href="rvsum_1.html#K16" title="RVSUM_1:func.16">Sum</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="flexary1.html#K4" title="FLEXARY1:func.4">|^</a> <font color="Maroon" title="c4">F1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rvsum_1.html#K16" title="RVSUM_1:func.16">Sum</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="flexary1.html#K4" title="FLEXARY1:func.4">|^</a> <span class="p2">(<span class="default"><a href="rfinseq2.html#K6" title="RFINSEQ2:func.6">sort_a</a> <font color="Maroon" title="c4">F1</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="rfinseq.html#T9" onmouseover="rs('rfinseq/T9')" onmouseout="rh()">RFINSEQ:9</a></span>;<br/>
<a NAME="E9:45"/><span class="lab"><font color="Green" title="E37">A9</font></span>: 
 <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <span class="p1">(<span class="default"><a href="rfinseq2.html#K6" title="RFINSEQ2:func.6">sort_a</a> <font color="Maroon" title="c5">F2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c3">f2</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="rfinseq2.html#D6" onmouseover="rs('rfinseq2/D6')" onmouseout="rh()">RFINSEQ2:def 6</a>, <a class="ref" href="classes1.html#T75" onmouseover="rs('classes1/T75')" onmouseout="rh()">CLASSES1:75</a></span>;<br/>
<a NAME="E10:45"/><span class="kw">then </span><span class="lab"><font color="Green" title="E38">A10</font></span>: 
 <a href="rfinseq2.html#K6" title="RFINSEQ2:func.6">sort_a</a> <font color="Maroon" title="c5">F2</font> is  <a href="valued_0.html#V6" title="VALUED_0:attr.6">natural-valued</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:45"><span class="lab"><font color="Green" title="E32">A2</font></span></a>, <a class="ref" href="valued_0.html#D6" onmouseover="rs('valued_0/D6')" onmouseout="rh()">VALUED_0:def 6</a></span>;<br/>
<a NAME="E11:45"/><span class="lab"><font color="Green" title="E39">A12</font></span>: 
 <a href="rfinseq2.html#K6" title="RFINSEQ2:func.6">sort_a</a> <font color="Maroon" title="c5">F2</font> is  <a href="valued_0.html#V6" title="VALUED_0:attr.6">natural-valued</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:45"><span class="lab"><font color="Green" title="E37">A9</font></span></a>, <a class="txt" href="#E2:45"><span class="lab"><font color="Green" title="E32">A2</font></span></a>, <a class="ref" href="valued_0.html#D6" onmouseover="rs('valued_0/D6')" onmouseout="rh()">VALUED_0:def 6</a></span>;<br/>
<a NAME="E12:45"/>
<font color="Maroon" title="c1">n</font> <a href="flexary1.html#K4" title="FLEXARY1:func.4">|^</a> <font color="Maroon" title="c5">F2</font>,<font color="Maroon" title="c1">n</font> <a href="flexary1.html#K4" title="FLEXARY1:func.4">|^</a> <span class="p1">(<span class="default"><a href="rfinseq2.html#K6" title="RFINSEQ2:func.6">sort_a</a> <font color="Maroon" title="c5">F2</font></span>)</span> <a href="classes1.html#R2" title="CLASSES1:pred.2">are_fiberwise_equipotent</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:45"><span class="lab"><font color="Green" title="E38">A10</font></span></a>, <a class="txt" href="#E4:45"><span class="lab"><font color="Green" title="E33">A3</font></span></a>, <a class="txt" href="#E1:45"><span class="lab"><font color="Green" title="E31">A1</font></span></a>, <a class="ref" href="flexary1.html#T29" target="_self" onmouseover="rs('flexary1/T29')" onmouseout="rh()">Th29</a></span>;<br/>
<a NAME="E13:45"/><span class="kw">then </span><span class="lab"><font color="Green" title="E40">A14</font></span>: 
 <a href="rvsum_1.html#K16" title="RVSUM_1:func.16">Sum</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="flexary1.html#K4" title="FLEXARY1:func.4">|^</a> <font color="Maroon" title="c5">F2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rvsum_1.html#K16" title="RVSUM_1:func.16">Sum</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="flexary1.html#K4" title="FLEXARY1:func.4">|^</a> <span class="p2">(<span class="default"><a href="rfinseq2.html#K6" title="RFINSEQ2:func.6">sort_a</a> <font color="Maroon" title="c5">F2</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="rfinseq.html#T9" onmouseover="rs('rfinseq/T9')" onmouseout="rh()">RFINSEQ:9</a></span>;<br/>
<a NAME="E14:45"/><span class="lab"><font color="Green" title="E41">A15</font></span>: 
 for <font color="Olive" title="b1">e1</font>, <font color="Olive" title="b2">e2</font> being   <a href="xxreal_0.html#NM1" title="XXREAL_0:NM.1">ExtReal</a>  st <font color="Olive" title="b1">e1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="rfinseq2.html#K6" title="RFINSEQ2:func.6">sort_a</a> <font color="Maroon" title="c4">F1</font></span>)</span> &amp; <font color="Olive" title="b2">e2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="rfinseq2.html#K6" title="RFINSEQ2:func.6">sort_a</a> <font color="Maroon" title="c4">F1</font></span>)</span> &amp; <font color="Olive" title="b1">e1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b2">e2</font> holds <br/><span class="p1">(<span class="default"><a href="rfinseq2.html#K6" title="RFINSEQ2:func.6">sort_a</a> <font color="Maroon" title="c4">F1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">e1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><a href="rfinseq2.html#K6" title="RFINSEQ2:func.6">sort_a</a> <font color="Maroon" title="c4">F1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">e2</font>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="45_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c8">e1</font>, <font color="Maroon" title="c9">e2</font> be   <a href="xxreal_0.html#NM1" title="XXREAL_0:NM.1">ExtReal</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="c8">e1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="rfinseq2.html#K6" title="RFINSEQ2:func.6">sort_a</a> <font color="Maroon" title="c4">F1</font></span>)</span> &amp; <font color="Maroon" title="c9">e2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="rfinseq2.html#K6" title="RFINSEQ2:func.6">sort_a</a> <font color="Maroon" title="c4">F1</font></span>)</span> &amp; <font color="Maroon" title="c8">e1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c9">e2</font> implies <span class="p1">(<span class="default"><a href="rfinseq2.html#K6" title="RFINSEQ2:func.6">sort_a</a> <font color="Maroon" title="c4">F1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">e1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><a href="rfinseq2.html#K6" title="RFINSEQ2:func.6">sort_a</a> <font color="Maroon" title="c4">F1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">e2</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:45_1"/><span class="lab"><font color="Green" title="E42">A16</font></span>: 
( <font color="Maroon" title="c8">e1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="rfinseq2.html#K6" title="RFINSEQ2:func.6">sort_a</a> <font color="Maroon" title="c4">F1</font></span>)</span> &amp; <font color="Maroon" title="c9">e2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="rfinseq2.html#K6" title="RFINSEQ2:func.6">sort_a</a> <font color="Maroon" title="c4">F1</font></span>)</span> &amp; <font color="Maroon" title="c8">e1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c9">e2</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"><a href="rfinseq2.html#K6" title="RFINSEQ2:func.6">sort_a</a> <font color="Maroon" title="c4">F1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">e1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><a href="rfinseq2.html#K6" title="RFINSEQ2:func.6">sort_a</a> <font color="Maroon" title="c4">F1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">e2</font></span><br/>

<a NAME="E2:45_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E43">A17</font></span>: 
<span class="p1">(<span class="default"><a href="rfinseq2.html#K6" title="RFINSEQ2:func.6">sort_a</a> <font color="Maroon" title="c4">F1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">e1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="rfinseq2.html#K6" title="RFINSEQ2:func.6">sort_a</a> <font color="Maroon" title="c4">F1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">e2</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="integra2.html#T2" onmouseover="rs('integra2/T2')" onmouseout="rh()">INTEGRA2:2</a></span>;<br/>
<span class="kw">assume </span><a NAME="E3:45_1"/><span class="lab"><font color="Green" title="E44">A18</font></span>: 
<span class="p1">(<span class="default"><a href="rfinseq2.html#K6" title="RFINSEQ2:func.6">sort_a</a> <font color="Maroon" title="c4">F1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">e1</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <span class="p1">(<span class="default"><a href="rfinseq2.html#K6" title="RFINSEQ2:func.6">sort_a</a> <font color="Maroon" title="c4">F1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">e2</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"> contradiction</span><br/>

<span class="kw">consider </span><font color="Maroon" title="c10">H</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><span class="kw"> such that </span><br/><a NAME="E5:45_1"/><span class="lab"><font color="Green" title="E45">A19</font></span>: 
(  <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <font color="Maroon" title="c10">H</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"><a href="rfinseq2.html#K6" title="RFINSEQ2:func.6">sort_a</a> <font color="Maroon" title="c4">F1</font></span>)</span> &amp;  <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <font color="Maroon" title="c10">H</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="c4">F1</font> &amp; <font color="Maroon" title="c10">H</font> is  <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>  &amp;  <a href="rfinseq2.html#K6" title="RFINSEQ2:func.6">sort_a</a> <font color="Maroon" title="c4">F1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">F1</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c10">H</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:45"><span class="lab"><font color="Green" title="E33">A3</font></span></a>, <a class="ref" href="classes1.html#T77" onmouseover="rs('classes1/T77')" onmouseout="rh()">CLASSES1:77</a></span>;<br/>
<a NAME="E6:45_1"/>
 <a href="rfinseq2.html#K6" title="RFINSEQ2:func.6">sort_a</a> <font color="Maroon" title="c4">F1</font> is  <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:45_1"><span class="lab"><font color="Green" title="E45">A19</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E7:45_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:45_1"><span class="lab"><font color="Green" title="E44">A18</font></span></a>, <a class="txt" href="#E2:45_1"><span class="lab"><font color="Green" title="E43">A17</font></span></a>, <a class="ref" href="xxreal_0.html#T1" onmouseover="rs('xxreal_0/T1')" onmouseout="rh()">XXREAL_0:1</a>, <a class="txt" href="#E1:45_1"><span class="lab"><font color="Green" title="E42">A16</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="E15:45"/>
 for <font color="Olive" title="b1">e1</font>, <font color="Olive" title="b2">e2</font> being   <a href="xxreal_0.html#NM1" title="XXREAL_0:NM.1">ExtReal</a>  st <font color="Olive" title="b1">e1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="rfinseq2.html#K6" title="RFINSEQ2:func.6">sort_a</a> <font color="Maroon" title="c5">F2</font></span>)</span> &amp; <font color="Olive" title="b2">e2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="rfinseq2.html#K6" title="RFINSEQ2:func.6">sort_a</a> <font color="Maroon" title="c5">F2</font></span>)</span> &amp; <font color="Olive" title="b1">e1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b2">e2</font> holds <br/><span class="p1">(<span class="default"><a href="rfinseq2.html#K6" title="RFINSEQ2:func.6">sort_a</a> <font color="Maroon" title="c5">F2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">e1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><a href="rfinseq2.html#K6" title="RFINSEQ2:func.6">sort_a</a> <font color="Maroon" title="c5">F2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">e2</font>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="45_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c8">e1</font>, <font color="Maroon" title="c9">e2</font> be   <a href="xxreal_0.html#NM1" title="XXREAL_0:NM.1">ExtReal</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="c8">e1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="rfinseq2.html#K6" title="RFINSEQ2:func.6">sort_a</a> <font color="Maroon" title="c5">F2</font></span>)</span> &amp; <font color="Maroon" title="c9">e2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="rfinseq2.html#K6" title="RFINSEQ2:func.6">sort_a</a> <font color="Maroon" title="c5">F2</font></span>)</span> &amp; <font color="Maroon" title="c8">e1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c9">e2</font> implies <span class="p1">(<span class="default"><a href="rfinseq2.html#K6" title="RFINSEQ2:func.6">sort_a</a> <font color="Maroon" title="c5">F2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">e1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><a href="rfinseq2.html#K6" title="RFINSEQ2:func.6">sort_a</a> <font color="Maroon" title="c5">F2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">e2</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:45_2"/><span class="lab"><font color="Green" title="E42">A20</font></span>: 
( <font color="Maroon" title="c8">e1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="rfinseq2.html#K6" title="RFINSEQ2:func.6">sort_a</a> <font color="Maroon" title="c5">F2</font></span>)</span> &amp; <font color="Maroon" title="c9">e2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="rfinseq2.html#K6" title="RFINSEQ2:func.6">sort_a</a> <font color="Maroon" title="c5">F2</font></span>)</span> &amp; <font color="Maroon" title="c8">e1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c9">e2</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"><a href="rfinseq2.html#K6" title="RFINSEQ2:func.6">sort_a</a> <font color="Maroon" title="c5">F2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">e1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><a href="rfinseq2.html#K6" title="RFINSEQ2:func.6">sort_a</a> <font color="Maroon" title="c5">F2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">e2</font></span><br/>

<a NAME="E2:45_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E43">A21</font></span>: 
<span class="p1">(<span class="default"><a href="rfinseq2.html#K6" title="RFINSEQ2:func.6">sort_a</a> <font color="Maroon" title="c5">F2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">e1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="rfinseq2.html#K6" title="RFINSEQ2:func.6">sort_a</a> <font color="Maroon" title="c5">F2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">e2</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="integra2.html#T2" onmouseover="rs('integra2/T2')" onmouseout="rh()">INTEGRA2:2</a></span>;<br/>
<span class="kw">assume </span><a NAME="E3:45_2"/><span class="lab"><font color="Green" title="E44">A22</font></span>: 
<span class="p1">(<span class="default"><a href="rfinseq2.html#K6" title="RFINSEQ2:func.6">sort_a</a> <font color="Maroon" title="c5">F2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">e1</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <span class="p1">(<span class="default"><a href="rfinseq2.html#K6" title="RFINSEQ2:func.6">sort_a</a> <font color="Maroon" title="c5">F2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">e2</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"> contradiction</span><br/>

<span class="kw">consider </span><font color="Maroon" title="c10">H</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><span class="kw"> such that </span><br/><a NAME="E5:45_2"/><span class="lab"><font color="Green" title="E45">A23</font></span>: 
(  <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <font color="Maroon" title="c10">H</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"><a href="rfinseq2.html#K6" title="RFINSEQ2:func.6">sort_a</a> <font color="Maroon" title="c5">F2</font></span>)</span> &amp;  <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <font color="Maroon" title="c10">H</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="c5">F2</font> &amp; <font color="Maroon" title="c10">H</font> is  <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>  &amp;  <a href="rfinseq2.html#K6" title="RFINSEQ2:func.6">sort_a</a> <font color="Maroon" title="c5">F2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">F2</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c10">H</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:45"><span class="lab"><font color="Green" title="E33">A3</font></span></a>, <a class="ref" href="classes1.html#T77" onmouseover="rs('classes1/T77')" onmouseout="rh()">CLASSES1:77</a></span>;<br/>
<a NAME="E6:45_2"/>
 <a href="rfinseq2.html#K6" title="RFINSEQ2:func.6">sort_a</a> <font color="Maroon" title="c5">F2</font> is  <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:45_2"><span class="lab"><font color="Green" title="E45">A23</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E7:45_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:45_2"><span class="lab"><font color="Green" title="E44">A22</font></span></a>, <a class="txt" href="#E2:45_2"><span class="lab"><font color="Green" title="E43">A21</font></span></a>, <a class="ref" href="xxreal_0.html#T1" onmouseover="rs('xxreal_0/T1')" onmouseout="rh()">XXREAL_0:1</a>, <a class="txt" href="#E1:45_2"><span class="lab"><font color="Green" title="E42">A20</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="E16:45"/><span class="kw">then </span><span class="lab"><font color="Green" title="E42">A24</font></span>: 
 <a href="rfinseq2.html#K6" title="RFINSEQ2:func.6">sort_a</a> <font color="Maroon" title="c5">F2</font> is  <a href="valued_0.html#V7" title="VALUED_0:attr.7">increasing</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="valued_0.html#D13" onmouseover="rs('valued_0/D13')" onmouseout="rh()">VALUED_0:def 13</a></span>;<br/>
<a NAME="E17:45"/><span class="lab"><font color="Green" title="E43">A25</font></span>: 
 <a href="rvsum_1.html#K16" title="RVSUM_1:func.16">Sum</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="flexary1.html#K4" title="FLEXARY1:func.4">|^</a> <span class="p2">(<span class="default"><a href="rfinseq2.html#K6" title="RFINSEQ2:func.6">sort_a</a> <font color="Maroon" title="c4">F1</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="flexary1.html#K4" title="FLEXARY1:func.4">|^</a> <span class="p3">(<span class="default"><a href="rfinseq2.html#K6" title="RFINSEQ2:func.6">sort_a</a> <font color="Maroon" title="c4">F1</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1</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"><font color="Maroon" title="c1">n</font> <a href="flexary1.html#K4" title="FLEXARY1:func.4">|^</a> <span class="p3">(<span class="default"><a href="rfinseq2.html#K6" title="RFINSEQ2:func.6">sort_a</a> <font color="Maroon" title="c4">F1</font></span>)</span></span>)</span>,2) <a href="flexary1.html#K3" title="FLEXARY1:func.3">+...</a></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="flexary1.html#T22" target="_self" onmouseover="rs('flexary1/T22')" onmouseout="rh()">Th22</a></span>;<br/>
<a NAME="E18:45"/>
 <a href="rvsum_1.html#K16" title="RVSUM_1:func.16">Sum</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="flexary1.html#K4" title="FLEXARY1:func.4">|^</a> <font color="Maroon" title="c2">f1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="flexary1.html#K4" title="FLEXARY1:func.4">|^</a> <font color="Maroon" title="c2">f1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1</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"><font color="Maroon" title="c1">n</font> <a href="flexary1.html#K4" title="FLEXARY1:func.4">|^</a> <font color="Maroon" title="c2">f1</font></span>)</span>,2) <a href="flexary1.html#K3" title="FLEXARY1:func.3">+...</a></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="flexary1.html#T22" target="_self" onmouseover="rs('flexary1/T22')" onmouseout="rh()">Th22</a></span>;<br/>
<a NAME="E19:45"/><span class="kw">then </span>
 <a href="rvsum_1.html#K16" title="RVSUM_1:func.16">Sum</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="flexary1.html#K4" title="FLEXARY1:func.4">|^</a> <span class="p2">(<span class="default"><a href="rfinseq2.html#K6" title="RFINSEQ2:func.6">sort_a</a> <font color="Maroon" title="c4">F1</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rvsum_1.html#K16" title="RVSUM_1:func.16">Sum</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="flexary1.html#K4" title="FLEXARY1:func.4">|^</a> <span class="p2">(<span class="default"><a href="rfinseq2.html#K6" title="RFINSEQ2:func.6">sort_a</a> <font color="Maroon" title="c5">F2</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="flexary1.html#T22" target="_self" onmouseover="rs('flexary1/T22')" onmouseout="rh()">Th22</a>, <a class="txt" href="#E1:45"><span class="lab"><font color="Green" title="E31">A1</font></span></a>, <a class="txt" href="#E8:45"><span class="lab"><font color="Green" title="E36">A8</font></span></a>, <a class="txt" href="#E13:45"><span class="lab"><font color="Green" title="E40">A14</font></span></a></span>;<br/>
<a NAME="E20:45"/><span class="kw">then </span>
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="flexary1.html#K4" title="FLEXARY1:func.4">|^</a> <span class="p3">(<span class="default"><a href="rfinseq2.html#K6" title="RFINSEQ2:func.6">sort_a</a> <font color="Maroon" title="c4">F1</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1</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"><font color="Maroon" title="c1">n</font> <a href="flexary1.html#K4" title="FLEXARY1:func.4">|^</a> <span class="p3">(<span class="default"><a href="rfinseq2.html#K6" title="RFINSEQ2:func.6">sort_a</a> <font color="Maroon" title="c4">F1</font></span>)</span></span>)</span>,2) <a href="flexary1.html#K3" title="FLEXARY1:func.3">+...</a></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="flexary1.html#K4" title="FLEXARY1:func.4">|^</a> <span class="p3">(<span class="default"><a href="rfinseq2.html#K6" title="RFINSEQ2:func.6">sort_a</a> <font color="Maroon" title="c5">F2</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1</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"><font color="Maroon" title="c1">n</font> <a href="flexary1.html#K4" title="FLEXARY1:func.4">|^</a> <span class="p3">(<span class="default"><a href="rfinseq2.html#K6" title="RFINSEQ2:func.6">sort_a</a> <font color="Maroon" title="c5">F2</font></span>)</span></span>)</span>,2) <a href="flexary1.html#K3" title="FLEXARY1:func.3">+...</a></span>)</span> &amp;  <a href="rfinseq2.html#K6" title="RFINSEQ2:func.6">sort_a</a> <font color="Maroon" title="c4">F1</font> is  <a href="valued_0.html#V7" title="VALUED_0:attr.7">increasing</a>  &amp;  <a href="rfinseq2.html#K6" title="RFINSEQ2:func.6">sort_a</a> <font color="Maroon" title="c4">F1</font> is  <a href="valued_0.html#V6" title="VALUED_0:attr.6">natural-valued</a>  )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:45"><span class="lab"><font color="Green" title="E41">A15</font></span></a>, <a class="ref" href="valued_0.html#D13" onmouseover="rs('valued_0/D13')" onmouseout="rh()">VALUED_0:def 13</a>, <a class="txt" href="#E17:45"><span class="lab"><font color="Green" title="E43">A25</font></span></a>, <a class="ref" href="flexary1.html#T22" target="_self" onmouseover="rs('flexary1/T22')" onmouseout="rh()">Th22</a>, <a class="txt" href="#E5:45"><span class="lab"><font color="Green" title="E34">A4</font></span></a>, <a class="txt" href="#E2:45"><span class="lab"><font color="Green" title="E32">A2</font></span></a>, <a class="ref" href="valued_0.html#D6" onmouseover="rs('valued_0/D6')" onmouseout="rh()">VALUED_0:def 6</a></span>;<br/>
<span class="kw">hence </span><a NAME="E21:45"/>
 <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c2">f1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c3">f2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:45"><span class="lab"><font color="Green" title="E31">A1</font></span></a>, <a class="txt" href="#E11:45"><span class="lab"><font color="Green" title="E39">A12</font></span></a>, <a class="txt" href="#E16:45"><span class="lab"><font color="Green" title="E42">A24</font></span></a>, <a class="ref" href="flexary1.html#T27" target="_self" onmouseover="rs('flexary1/T27')" onmouseout="rh()">Th27</a>, <a class="txt" href="#E9:45"><span class="lab"><font color="Green" title="E37">A9</font></span></a>, <a class="txt" href="#E5:45"><span class="lab"><font color="Green" title="E34">A4</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>
