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<b>let </b><font color="Maroon" title="c1">r</font> be  <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> ;<br/><b>let </b><font color="Maroon" title="c2">f</font> be  <a href="seq_1.html#V1" title="SEQ_1:attr.1">real-yielding</a> <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a>;<br/>



<b>set </b><font color="Maroon" title="c3">n</font> =  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">f</font>;<br/>
<b>reconsider </b><font color="Maroon" title="c4">x</font> = <font color="Maroon" title="c2">f</font> as    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of  <a href="euclid.html#K1" title="EUCLID:func.1">REAL</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">f</font></span>)</span> <b>by </b><i><a class="txt" href="euclid.html#E13"><i><font color="Green" title="E5">Lm2</font></i></a></i>;<br/>
<a NAME="E2:25"/><i><font color="Green" title="E14">A1</font></i>: 
( 0 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <font color="Maroon" title="c1">r</font></span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a>  &amp; 0 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K13" title="RVSUM_1:func.13">sqr</a> <span class="p2">(<span class="default"><a href="euclid.html#K10" title="EUCLID:func.10">abs</a> <font color="Maroon" title="c4">x</font></span>)</span></span>)</span> )
 
<b>by </b><i><a class="ref" href="rvsum_1.html#T116" title="RVSUM_1:th.116">RVSUM_1:116</a>, <a class="ref" href="xreal_1.html#T65" title="XREAL_1:th.65">XREAL_1:65</a></i>;<br/>
<a NAME="E3:25"/><i><font color="Green" title="E15">A2</font></i>: 
0 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <font color="Maroon" title="c1">r</font>
 
<b>by </b><i><a class="ref" href="complex1.html#T132" title="COMPLEX1:th.132">COMPLEX1:132</a></i>;<br/>
<b>thus </b><span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">r</font> <a href="rvsum_1.html#K10" title="RVSUM_1:func.10">*</a> <font color="Maroon" title="c2">f</font></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> = 
 <a href="square_1.html#K7" title="SQUARE_1:func.7">sqrt</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p2">(<span class="default"><a href="rvsum_1.html#K13" title="RVSUM_1:func.13">sqr</a> <span class="p3">(<span class="default"><a href="euclid.html#K10" title="EUCLID:func.10">abs</a> <span class="p4">(<span class="default"><font color="Maroon" title="c1">r</font> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <font color="Maroon" title="c4">x</font></span>)</span></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="euclid.html#E18"><i><font color="Green" title="E9">Lm3</font></i></a></i>
<br/>.= 
 <a href="square_1.html#K7" title="SQUARE_1:func.7">sqrt</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p2">(<span class="default"><a href="rvsum_1.html#K13" title="RVSUM_1:func.13">sqr</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <font color="Maroon" title="c1">r</font></span>)</span> <a href="rvsum_1.html#K11" title="RVSUM_1:func.11">*</a> <span class="p4">(<span class="default"><a href="euclid.html#K10" title="EUCLID:func.10">abs</a> <font color="Maroon" title="c4">x</font></span>)</span></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="euclid.html#T9" target="_self" title="EUCLID:th.9">Th9</a></i>
<br/>.= 
 <a href="square_1.html#K7" title="SQUARE_1:func.7">sqrt</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <font color="Maroon" title="c1">r</font></span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span> <a href="rvsum_1.html#K11" title="RVSUM_1:func.11">*</a> <span class="p3">(<span class="default"><a href="rvsum_1.html#K13" title="RVSUM_1:func.13">sqr</a> <span class="p4">(<span class="default"><a href="euclid.html#K10" title="EUCLID:func.10">abs</a> <font color="Maroon" title="c4">x</font></span>)</span></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="rvsum_1.html#T84" title="RVSUM_1:th.84">RVSUM_1:84</a></i>
<br/>.= 
 <a href="square_1.html#K7" title="SQUARE_1:func.7">sqrt</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <font color="Maroon" title="c1">r</font></span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <span class="p2">(<span class="default"><a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p3">(<span class="default"><a href="rvsum_1.html#K13" title="RVSUM_1:func.13">sqr</a> <span class="p4">(<span class="default"><a href="euclid.html#K10" title="EUCLID:func.10">abs</a> <font color="Maroon" title="c4">x</font></span>)</span></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="rvsum_1.html#T117" title="RVSUM_1:th.117">RVSUM_1:117</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="square_1.html#K7" title="SQUARE_1:func.7">sqrt</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <font color="Maroon" title="c1">r</font></span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span></span>)</span> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <span class="p1">(<span class="default"><a href="square_1.html#K7" title="SQUARE_1:func.7">sqrt</a> <span class="p2">(<span class="default"><a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p3">(<span class="default"><a href="rvsum_1.html#K13" title="RVSUM_1:func.13">sqr</a> <span class="p4">(<span class="default"><a href="euclid.html#K10" title="EUCLID:func.10">abs</a> <font color="Maroon" title="c4">x</font></span>)</span></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="euclid.html#E2:25"><i><font color="Green" title="E14">A1</font></i></a>, <a class="ref" href="square_1.html#T97" title="SQUARE_1:th.97">SQUARE_1:97</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <font color="Maroon" title="c1">r</font></span>)</span> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <span class="p1">(<span class="default"><a href="square_1.html#K7" title="SQUARE_1:func.7">sqrt</a> <span class="p2">(<span class="default"><a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p3">(<span class="default"><a href="rvsum_1.html#K13" title="RVSUM_1:func.13">sqr</a> <span class="p4">(<span class="default"><a href="euclid.html#K10" title="EUCLID:func.10">abs</a> <font color="Maroon" title="c4">x</font></span>)</span></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="euclid.html#E3:25"><i><font color="Green" title="E15">A2</font></i></a>, <a class="ref" href="square_1.html#T89" title="SQUARE_1:th.89">SQUARE_1:89</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <font color="Maroon" title="c1">r</font></span>)</span> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><font color="Maroon" title="c2">f</font></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span>
<b>by </b><i><a class="txt" href="euclid.html#E18"><i><font color="Green" title="E9">Lm3</font></i></a></i>
;<br/>


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