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<b>let </b><font color="Maroon" title="c2">a</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="c3">r</font> be  non <a href="xxreal_0.html#V3" title="XXREAL_0:attr.3">negative</a> <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="c4">n</font> be  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/><b>let </b><font color="Maroon" title="c5">s</font>, <font color="Maroon" title="c6">t</font>, <font color="Maroon" title="c7">o</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c4">n</font></span>)</span>;<br/><b>let </b><font color="Maroon" title="c8">S</font>, <font color="Maroon" title="c9">T</font>, <font color="Maroon" title="c10">O</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="euclid.html#K1" title="EUCLID:func.1">REAL</a> <font color="Maroon" title="c4">n</font>;<br/>













<b>assume </b><b>that </b><br/><a NAME="E1:13"/><i><font color="Green" title="E7">A1</font></i>: 
( <font color="Maroon" title="c8">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">s</font> &amp; <font color="Maroon" title="c9">T</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">t</font> &amp; <font color="Maroon" title="c10">O</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">o</font> )
 <b>and </b><br/><a NAME="E2:13"/><i><font color="Green" title="E8">A2</font></i>: 
<font color="Maroon" title="c5">s</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="brouwer.html#K2" title="BROUWER:func.2">Tdisk</a> <font color="Maroon" title="c7">o</font>,<font color="Maroon" title="c3">r</font></span>)</span>
 <b>and </b><br/><a NAME="E3:13"/><i><font color="Green" title="E9">A3</font></i>: 
<font color="Maroon" title="c6">t</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="brouwer.html#K2" title="BROUWER:func.2">Tdisk</a> <font color="Maroon" title="c7">o</font>,<font color="Maroon" title="c3">r</font></span>)</span>
 <b>and </b><br/><a NAME="E4:13"/><i><font color="Green" title="E10">A4</font></i>: 
<font color="Maroon" title="c5">s</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c6">t</font>
 ;<br/>

<b>set </b><font color="Maroon" title="c11">H</font> =  <a href="brouwer.html#K3" title="BROUWER:func.3">HC</a> <font color="Maroon" title="c5">s</font>,<font color="Maroon" title="c6">t</font>,<font color="Maroon" title="c7">o</font>,<font color="Maroon" title="c3">r</font>;<br/>
<b>set </b><font color="Maroon" title="c12">M</font> = <span class="p1"><a href="euclid_2.html#K2" title="EUCLID_2:func.2">|(</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">t</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c5">s</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c5">s</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c7">o</font></span>)</span></span><a href="euclid_2.html#K2" title="EUCLID_2:func.2">)|</a></span>;<br/>
<b>set </b><font color="Maroon" title="c13">A</font> =  <a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p1">(<span class="default"><a href="euclid.html#K11" title="EUCLID:func.11">sqr</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">T</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c8">S</font></span>)</span></span>)</span>;<br/>
<b>set </b><font color="Maroon" title="c14">B</font> = 2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1"><a href="euclid_2.html#K2" title="EUCLID_2:func.2">|(</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">t</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c5">s</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c5">s</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c7">o</font></span>)</span></span><a href="euclid_2.html#K2" title="EUCLID_2:func.2">)|</a></span>;<br/>
<b>set </b><font color="Maroon" title="c15">C</font> = <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="euclid.html#K11" title="EUCLID:func.11">sqr</a> <span class="p3">(<span class="default"><font color="Maroon" title="c8">S</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c10">O</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">r</font> <a href="square_1.html#K3" title="SQUARE_1:func.3">^2</a> </span>)</span>;<br/>
<b>set </b><font color="Maroon" title="c16">l1</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p3">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p4"><a href="euclid_2.html#K2" title="EUCLID_2:func.2">|(</a><span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c6">t</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c5">s</font></span>)</span>,<span class="p5">(<span class="default"><font color="Maroon" title="c5">s</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c7">o</font></span>)</span></span><a href="euclid_2.html#K2" title="EUCLID_2:func.2">)|</a></span></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default"><a href="square_1.html#K7" title="SQUARE_1:func.7">sqrt</a> <span class="p3">(<span class="default"><a href="quin_1.html#K2" title="QUIN_1:func.2">delta</a> <span class="p4">(<span class="default"><a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p5">(<span class="default"><a href="euclid.html#K11" title="EUCLID:func.11">sqr</a> <span class="p0">(<span class="default"><font color="Maroon" title="c9">T</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c8">S</font></span>)</span></span>)</span></span>)</span>,<span class="p4">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p5"><a href="euclid_2.html#K2" title="EUCLID_2:func.2">|(</a><span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c6">t</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c5">s</font></span>)</span>,<span class="p0">(<span class="default"><font color="Maroon" title="c5">s</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c7">o</font></span>)</span></span><a href="euclid_2.html#K2" title="EUCLID_2:func.2">)|</a></span></span>)</span>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p0">(<span class="default"><a href="euclid.html#K11" title="EUCLID:func.11">sqr</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">S</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c10">O</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p5">(<span class="default"><font color="Maroon" title="c3">r</font> <a href="square_1.html#K3" title="SQUARE_1:func.3">^2</a> </span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</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="euclid.html#K11" title="EUCLID:func.11">sqr</a> <span class="p4">(<span class="default"><font color="Maroon" title="c9">T</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c8">S</font></span>)</span></span>)</span></span>)</span></span>)</span>;<br/>
<b>set </b><font color="Maroon" title="c17">l2</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p3">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p4"><a href="euclid_2.html#K2" title="EUCLID_2:func.2">|(</a><span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c6">t</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c5">s</font></span>)</span>,<span class="p5">(<span class="default"><font color="Maroon" title="c5">s</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c7">o</font></span>)</span></span><a href="euclid_2.html#K2" title="EUCLID_2:func.2">)|</a></span></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p2">(<span class="default"><a href="square_1.html#K7" title="SQUARE_1:func.7">sqrt</a> <span class="p3">(<span class="default"><a href="quin_1.html#K2" title="QUIN_1:func.2">delta</a> <span class="p4">(<span class="default"><a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p5">(<span class="default"><a href="euclid.html#K11" title="EUCLID:func.11">sqr</a> <span class="p0">(<span class="default"><font color="Maroon" title="c9">T</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c8">S</font></span>)</span></span>)</span></span>)</span>,<span class="p4">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p5"><a href="euclid_2.html#K2" title="EUCLID_2:func.2">|(</a><span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c6">t</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c5">s</font></span>)</span>,<span class="p0">(<span class="default"><font color="Maroon" title="c5">s</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c7">o</font></span>)</span></span><a href="euclid_2.html#K2" title="EUCLID_2:func.2">)|</a></span></span>)</span>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p0">(<span class="default"><a href="euclid.html#K11" title="EUCLID:func.11">sqr</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">S</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c10">O</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p5">(<span class="default"><font color="Maroon" title="c3">r</font> <a href="square_1.html#K3" title="SQUARE_1:func.3">^2</a> </span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</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="euclid.html#K11" title="EUCLID:func.11">sqr</a> <span class="p4">(<span class="default"><font color="Maroon" title="c9">T</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c8">S</font></span>)</span></span>)</span></span>)</span></span>)</span>;<br/>
<b>assume </b><a NAME="E5:13"/><i><font color="Green" title="E11">A5</font></i>: 
<font color="Maroon" title="c2">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p3"><a href="euclid_2.html#K2" title="EUCLID_2:func.2">|(</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c6">t</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c5">s</font></span>)</span>,<span class="p4">(<span class="default"><font color="Maroon" title="c5">s</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c7">o</font></span>)</span></span><a href="euclid_2.html#K2" title="EUCLID_2:func.2">)|</a></span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p2">(<span class="default"><a href="square_1.html#K7" title="SQUARE_1:func.7">sqrt</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5"><a href="euclid_2.html#K2" title="EUCLID_2:func.2">|(</a><span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c6">t</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c5">s</font></span>)</span>,<span class="p0">(<span class="default"><font color="Maroon" title="c5">s</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c7">o</font></span>)</span></span><a href="euclid_2.html#K2" title="EUCLID_2:func.2">)|</a></span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p0">(<span class="default"><a href="euclid.html#K11" title="EUCLID:func.11">sqr</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">T</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c8">S</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p1">(<span class="default"><a href="euclid.html#K11" title="EUCLID:func.11">sqr</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">S</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c10">O</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p0">(<span class="default"><font color="Maroon" title="c3">r</font> <a href="square_1.html#K3" title="SQUARE_1:func.3">^2</a> </span>)</span></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</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="euclid.html#K11" title="EUCLID:func.11">sqr</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">T</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c8">S</font></span>)</span></span>)</span></span>)</span>
 ;<br/>

<a NAME="E6:13"/><i><font color="Green" title="E12">A6</font></i>: 
 <a href="brouwer.html#K3" title="BROUWER:func.3">HC</a> <font color="Maroon" title="c5">s</font>,<font color="Maroon" title="c6">t</font>,<font color="Maroon" title="c7">o</font>,<font color="Maroon" title="c3">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="topreal9.html#K4" title="TOPREAL9:func.4">halfline</a> <font color="Maroon" title="c5">s</font>,<font color="Maroon" title="c6">t</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal9.html#K3" title="TOPREAL9:func.3">Sphere</a> <font color="Maroon" title="c7">o</font>,<font color="Maroon" title="c3">r</font></span>)</span>
 
<b>by </b><i><a class="txt" href="brouwer.html#E2:13"><i><font color="Green" title="E8">A2</font></i></a>, <a class="txt" href="brouwer.html#E3:13"><i><font color="Green" title="E9">A3</font></i></a>, <a class="txt" href="brouwer.html#E4:13"><i><font color="Green" title="E10">A4</font></i></a>, <a class="ref" href="brouwer.html#D3" target="_self" title="BROUWER:def.3">Def3</a></i>;<br/>
<a NAME="E7:13"/><b>then </b>
 <a href="brouwer.html#K3" title="BROUWER:func.3">HC</a> <font color="Maroon" title="c5">s</font>,<font color="Maroon" title="c6">t</font>,<font color="Maroon" title="c7">o</font>,<font color="Maroon" title="c3">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal9.html#K4" title="TOPREAL9:func.4">halfline</a> <font color="Maroon" title="c5">s</font>,<font color="Maroon" title="c6">t</font>
 
<b>by </b><i><a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c18">l</font> being  <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> <b> such that </b><br/><a NAME="E9:13"/><i><font color="Green" title="E13">A7</font></i>: 
 <a href="brouwer.html#K3" title="BROUWER:func.3">HC</a> <font color="Maroon" title="c5">s</font>,<font color="Maroon" title="c6">t</font>,<font color="Maroon" title="c7">o</font>,<font color="Maroon" title="c3">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c18">l</font></span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <font color="Maroon" title="c5">s</font></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c18">l</font> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <font color="Maroon" title="c6">t</font></span>)</span>
 <b>and </b><br/><a NAME="E10:13"/><i><font color="Green" title="E14">A8</font></i>: 
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c18">l</font>
 <b>by </b><i><a class="ref" href="topreal9.html#T26" title="TOPREAL9:th.26">TOPREAL9:26</a></i>;<br/>
<i><font color="Green" title="E15">A9</font></i>:  <a href="brouwer.html#K3" title="BROUWER:func.3">HC</a> <font color="Maroon" title="c5">s</font>,<font color="Maroon" title="c6">t</font>,<font color="Maroon" title="c7">o</font>,<font color="Maroon" title="c3">r</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <font color="Maroon" title="c5">s</font></span>)</span> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c18">l</font> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <font color="Maroon" title="c5">s</font></span>)</span></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c18">l</font> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <font color="Maroon" title="c6">t</font></span>)</span>
<b>by </b><i><a class="txt" href="brouwer.html#E9:13"><i><font color="Green" title="E13">A7</font></i></a>, <a class="ref" href="euclid.html#T54" title="EUCLID:th.54">EUCLID:54</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">s</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c18">l</font> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <font color="Maroon" title="c5">s</font></span>)</span></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c18">l</font> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <font color="Maroon" title="c6">t</font></span>)</span>
<b>by </b><i><a class="ref" href="euclid.html#T33" title="EUCLID:th.33">EUCLID:33</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">s</font> <a href="euclid.html#K17" title="EUCLID:func.17">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c18">l</font> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <font color="Maroon" title="c6">t</font></span>)</span></span>)</span> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c18">l</font> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <font color="Maroon" title="c5">s</font></span>)</span>
<b>by </b><i><a class="ref" href="jordan2c.html#T9" title="JORDAN2C:th.9">JORDAN2C:9</a></i>
<br/>.= 
<font color="Maroon" title="c5">s</font> <a href="euclid.html#K17" title="EUCLID:func.17">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c18">l</font> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <font color="Maroon" title="c6">t</font></span>)</span> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c18">l</font> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <font color="Maroon" title="c5">s</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="euclid.html#T49" title="EUCLID:th.49">EUCLID:49</a></i>
<br/>.= 
<font color="Maroon" title="c5">s</font> <a href="euclid.html#K17" title="EUCLID:func.17">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c18">l</font> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">t</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c5">s</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="euclid.html#T53" title="EUCLID:th.53">EUCLID:53</a></i>
;<br/>
<a NAME="E12:13"/><i><font color="Green" title="E16">A10</font></i>: 
<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="c9">T</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c8">S</font></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<b>by </b><i><a class="txt" href="brouwer.html#E1:13"><i><font color="Green" title="E7">A1</font></i></a>, <a class="txt" href="brouwer.html#E4:13"><i><font color="Green" title="E10">A4</font></i></a>, <a class="ref" href="euclid.html#T19" title="EUCLID:th.19">EUCLID:19</a></i>;<br/>
<div><i><font color="Green" title="E17">A11</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="13_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:13_2"/>
 <a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p1">(<span class="default"><a href="euclid.html#K11" title="EUCLID:func.11">sqr</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">T</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c8">S</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ;<br/><a NAME="E2:13_2"/><b>then </b>
 <a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p1">(<span class="default"><a href="euclid.html#K11" title="EUCLID:func.11">sqr</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">T</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c8">S</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>by </b><i><a class="ref" href="rvsum_1.html#T116" title="RVSUM_1:th.116">RVSUM_1:116</a></i>;<br/><b>hence </b><a NAME="E3:13_2"/>
contradiction
 <b>by </b><i><a class="txt" href="brouwer.html#E12:13"><i><font color="Green" title="E16">A10</font></i></a>, <a class="ref" href="euclid.html#D5" title="EUCLID:def.5">EUCLID:def 5</a>, <a class="ref" href="square_1.html#T82" title="SQUARE_1:th.82">SQUARE_1:82</a></i>;<br/></div><b>end;</b></div>
<i><font color="Green" title="E18">A12</font></i>:  <a href="quin_1.html#K2" title="QUIN_1:func.2">delta</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="euclid.html#K11" title="EUCLID:func.11">sqr</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">T</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c8">S</font></span>)</span></span>)</span></span>)</span>,<span class="p1">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2"><a href="euclid_2.html#K2" title="EUCLID_2:func.2">|(</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c6">t</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c5">s</font></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon" title="c5">s</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c7">o</font></span>)</span></span><a href="euclid_2.html#K2" title="EUCLID_2:func.2">)|</a></span></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p3">(<span class="default"><a href="euclid.html#K11" title="EUCLID:func.11">sqr</a> <span class="p4">(<span class="default"><font color="Maroon" title="c8">S</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c10">O</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">r</font> <a href="square_1.html#K3" title="SQUARE_1:func.3">^2</a> </span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p3"><a href="euclid_2.html#K2" title="EUCLID_2:func.2">|(</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c6">t</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c5">s</font></span>)</span>,<span class="p4">(<span class="default"><font color="Maroon" title="c5">s</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c7">o</font></span>)</span></span><a href="euclid_2.html#K2" title="EUCLID_2:func.2">)|</a></span></span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">4 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p3">(<span class="default"><a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p4">(<span class="default"><a href="euclid.html#K11" title="EUCLID:func.11">sqr</a> <span class="p5">(<span class="default"><font color="Maroon" title="c9">T</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c8">S</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p4">(<span class="default"><a href="euclid.html#K11" title="EUCLID:func.11">sqr</a> <span class="p5">(<span class="default"><font color="Maroon" title="c8">S</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c10">O</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">r</font> <a href="square_1.html#K3" title="SQUARE_1:func.3">^2</a> </span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="quin_1.html#D1" title="QUIN_1:def.1">QUIN_1:def 1</a></i>
<br/>.= 
4 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="euclid_2.html#K2" title="EUCLID_2:func.2">|(</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c6">t</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c5">s</font></span>)</span>,<span class="p4">(<span class="default"><font color="Maroon" title="c5">s</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c7">o</font></span>)</span></span><a href="euclid_2.html#K2" title="EUCLID_2:func.2">)|</a></span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p4">(<span class="default"><a href="euclid.html#K11" title="EUCLID:func.11">sqr</a> <span class="p5">(<span class="default"><font color="Maroon" title="c9">T</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c8">S</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p5">(<span class="default"><a href="euclid.html#K11" title="EUCLID:func.11">sqr</a> <span class="p0">(<span class="default"><font color="Maroon" title="c8">S</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c10">O</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">r</font> <a href="square_1.html#K3" title="SQUARE_1:func.3">^2</a> </span>)</span></span>)</span></span>)</span></span>)</span>

;<br/>
<a NAME="E15:13"/>
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="brouwer.html#K2" title="BROUWER:func.2">Tdisk</a> <font color="Maroon" title="c7">o</font>,<font color="Maroon" title="c3">r</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="topreal9.html#K2" title="TOPREAL9:func.2">cl_Ball</a> <font color="Maroon" title="c7">o</font>,<font color="Maroon" title="c3">r</font>
 
<b>by </b><i><a class="ref" href="brouwer.html#T3" target="_self" title="BROUWER:th.3">Th3</a></i>;<br/>
<a NAME="E16:13"/><b>then </b><i><font color="Green" title="E19">A13</font></i>: 
<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="c5">s</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c7">o</font></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">r</font>
 
<b>by </b><i><a class="txt" href="brouwer.html#E2:13"><i><font color="Green" title="E8">A2</font></i></a>, <a class="ref" href="topreal9.html#T8" title="TOPREAL9:th.8">TOPREAL9:8</a></i>;<br/>
<a NAME="E17:13"/><i><font color="Green" title="E20">A14</font></i>: 
<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="c8">S</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c10">O</font></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <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="euclid.html#K11" title="EUCLID:func.11">sqr</a> <span class="p3">(<span class="default"><font color="Maroon" title="c8">S</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c10">O</font></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="euclid.html#D5" title="EUCLID:def.5">EUCLID:def 5</a></i>;<br/>
<a NAME="E18:13"/><i><font color="Green" title="E21">A15</font></i>: 
<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="c9">T</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c8">S</font></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <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="euclid.html#K11" title="EUCLID:func.11">sqr</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">T</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c8">S</font></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="euclid.html#D5" title="EUCLID:def.5">EUCLID:def 5</a></i>;<br/>
<a NAME="E19:13"/>
 <a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p1">(<span class="default"><a href="euclid.html#K11" title="EUCLID:func.11">sqr</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">T</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c8">S</font></span>)</span></span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<b>by </b><i><a class="ref" href="rvsum_1.html#T116" title="RVSUM_1:th.116">RVSUM_1:116</a></i>;<br/>
<a NAME="E20:13"/><b>then </b><i><font color="Green" title="E22">A16</font></i>: 
<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="c9">T</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c8">S</font></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a>  <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"><a href="euclid.html#K11" title="EUCLID:func.11">sqr</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">T</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c8">S</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="brouwer.html#E18:13"><i><font color="Green" title="E21">A15</font></i></a>, <a class="ref" href="square_1.html#D4" title="SQUARE_1:def.4">SQUARE_1:def 4</a></i>;<br/>
<a NAME="E21:13"/>
 <a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p1">(<span class="default"><a href="euclid.html#K11" title="EUCLID:func.11">sqr</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">S</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c10">O</font></span>)</span></span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<b>by </b><i><a class="ref" href="rvsum_1.html#T116" title="RVSUM_1:th.116">RVSUM_1:116</a></i>;<br/>
<a NAME="E22:13"/><b>then </b><i><font color="Green" title="E23">A17</font></i>: 
<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="c8">S</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c10">O</font></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a>  <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"><a href="euclid.html#K11" title="EUCLID:func.11">sqr</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">S</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c10">O</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="brouwer.html#E17:13"><i><font color="Green" title="E20">A14</font></i></a>, <a class="ref" href="square_1.html#D4" title="SQUARE_1:def.4">SQUARE_1:def 4</a></i>;<br/>
<a NAME="E23:13"/><i><font color="Green" title="E24">A18</font></i>: 
<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="euclid.html#K11" title="EUCLID:func.11">sqr</a> <span class="p4">(<span class="default"><font color="Maroon" title="c8">S</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c10">O</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">r</font> <a href="square_1.html#K3" title="SQUARE_1:func.3">^2</a> 
 
<b>by </b><i><a class="txt" href="brouwer.html#E1:13"><i><font color="Green" title="E7">A1</font></i></a>, <a class="txt" href="brouwer.html#E16:13"><i><font color="Green" title="E19">A13</font></i></a>, <a class="txt" href="brouwer.html#E17:13"><i><font color="Green" title="E20">A14</font></i></a>, <a class="ref" href="square_1.html#T77" title="SQUARE_1:th.77">SQUARE_1:77</a></i>;<br/>
<a NAME="E24:13"/><b>then </b><i><font color="Green" title="E25">A19</font></i>: 
<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="euclid.html#K11" title="EUCLID:func.11">sqr</a> <span class="p3">(<span class="default"><font color="Maroon" title="c8">S</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c10">O</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">r</font> <a href="square_1.html#K3" title="SQUARE_1:func.3">^2</a> </span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<b>by </b><i><a class="txt" href="brouwer.html#E17:13"><i><font color="Green" title="E20">A14</font></i></a>, <a class="txt" href="brouwer.html#E22:13"><i><font color="Green" title="E23">A17</font></i></a>, <a class="ref" href="real_2.html#T106" title="REAL_2:th.106">REAL_2:106</a></i>;<br/>
<a NAME="E25:13"/><i><font color="Green" title="E26">A20</font></i>: 
 <a href="quin_1.html#K2" title="QUIN_1:func.2">delta</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="euclid.html#K11" title="EUCLID:func.11">sqr</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">T</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c8">S</font></span>)</span></span>)</span></span>)</span>,<span class="p1">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2"><a href="euclid_2.html#K2" title="EUCLID_2:func.2">|(</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c6">t</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c5">s</font></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon" title="c5">s</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c7">o</font></span>)</span></span><a href="euclid_2.html#K2" title="EUCLID_2:func.2">)|</a></span></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p3">(<span class="default"><a href="euclid.html#K11" title="EUCLID:func.11">sqr</a> <span class="p4">(<span class="default"><font color="Maroon" title="c8">S</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c10">O</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">r</font> <a href="square_1.html#K3" title="SQUARE_1:func.3">^2</a> </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">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p3"><a href="euclid_2.html#K2" title="EUCLID_2:func.2">|(</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c6">t</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c5">s</font></span>)</span>,<span class="p4">(<span class="default"><font color="Maroon" title="c5">s</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c7">o</font></span>)</span></span><a href="euclid_2.html#K2" title="EUCLID_2:func.2">)|</a></span></span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">4 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p3">(<span class="default"><a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p4">(<span class="default"><a href="euclid.html#K11" title="EUCLID:func.11">sqr</a> <span class="p5">(<span class="default"><font color="Maroon" title="c9">T</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c8">S</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p4">(<span class="default"><a href="euclid.html#K11" title="EUCLID:func.11">sqr</a> <span class="p5">(<span class="default"><font color="Maroon" title="c8">S</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c10">O</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">r</font> <a href="square_1.html#K3" title="SQUARE_1:func.3">^2</a> </span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="quin_1.html#D1" title="QUIN_1:def.1">QUIN_1:def 1</a></i>;<br/>
<a NAME="E26:13"/>
4 <a href="real_1.html#K8" title="REAL_1:func.8">*</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="euclid.html#K11" title="EUCLID:func.11">sqr</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">T</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c8">S</font></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<b>by </b><i><a class="txt" href="brouwer.html#E13:13"><i><font color="Green" title="E17">A11</font></i></a>, <a class="ref" href="xreal_1.html#T131" title="XREAL_1:th.131">XREAL_1:131</a></i>;<br/>
<a NAME="E27:13"/><b>then </b>
<span class="p1">(<span class="default">4 <a href="real_1.html#K8" title="REAL_1:func.8">*</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="euclid.html#K11" title="EUCLID:func.11">sqr</a> <span class="p4">(<span class="default"><font color="Maroon" title="c9">T</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c8">S</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p3">(<span class="default"><a href="euclid.html#K11" title="EUCLID:func.11">sqr</a> <span class="p4">(<span class="default"><font color="Maroon" title="c8">S</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c10">O</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">r</font> <a href="square_1.html#K3" title="SQUARE_1:func.3">^2</a> </span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<b>by </b><i><a class="txt" href="brouwer.html#E24:13"><i><font color="Green" title="E25">A19</font></i></a>, <a class="ref" href="xreal_1.html#T133" title="XREAL_1:th.133">XREAL_1:133</a></i>;<br/>
<a NAME="E28:13"/><b>then </b><i><font color="Green" title="E27">A21</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p3"><a href="euclid_2.html#K2" title="EUCLID_2:func.2">|(</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c6">t</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c5">s</font></span>)</span>,<span class="p4">(<span class="default"><font color="Maroon" title="c5">s</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c7">o</font></span>)</span></span><a href="euclid_2.html#K2" title="EUCLID_2:func.2">)|</a></span></span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">4 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p3">(<span class="default"><a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p4">(<span class="default"><a href="euclid.html#K11" title="EUCLID:func.11">sqr</a> <span class="p5">(<span class="default"><font color="Maroon" title="c9">T</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c8">S</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p4">(<span class="default"><a href="euclid.html#K11" title="EUCLID:func.11">sqr</a> <span class="p5">(<span class="default"><font color="Maroon" title="c8">S</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c10">O</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">r</font> <a href="square_1.html#K3" title="SQUARE_1:func.3">^2</a> </span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<b>by </b><i><a class="ref" href="real_2.html#T174" title="REAL_2:th.174">REAL_2:174</a></i>;<br/>
<a NAME="E29:13"/>
<font color="Maroon" title="c18">l</font> is   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>
 
<b>by </b><i><a class="ref" href="xreal_0.html#D1" title="XREAL_0:def.1">XREAL_0:def 1</a></i>;<br/>
<b>then </b><i><font color="Green" title="E28">A22</font></i>: <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="c18">l</font> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c6">t</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c5">s</font></span>)</span></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a>  = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <font color="Maroon" title="c18">l</font></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c6">t</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c5">s</font></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span></span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> 
<b>by </b><i><a class="ref" href="toprns_1.html#T8" title="TOPRNS_1:th.8">TOPRNS_1:8</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <font color="Maroon" title="c18">l</font></span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><span class="p2"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c6">t</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c5">s</font></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span>

<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c18">l</font> <a href="square_1.html#K3" title="SQUARE_1:func.3">^2</a> </span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p1">(<span class="default"><span class="p2"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c6">t</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c5">s</font></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span>
<b>by </b><i><a class="ref" href="complex1.html#T161" title="COMPLEX1:th.161">COMPLEX1:161</a></i>
;<br/>
<a NAME="E31:13"/>
 <a href="brouwer.html#K3" title="BROUWER:func.3">HC</a> <font color="Maroon" title="c5">s</font>,<font color="Maroon" title="c6">t</font>,<font color="Maroon" title="c7">o</font>,<font color="Maroon" title="c3">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal9.html#K3" title="TOPREAL9:func.3">Sphere</a> <font color="Maroon" title="c7">o</font>,<font color="Maroon" title="c3">r</font>
 
<b>by </b><i><a class="txt" href="brouwer.html#E6:13"><i><font color="Green" title="E12">A6</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/>
<b>then </b><font color="Maroon" title="c3">r</font> = 
<span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default">1 <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c18">l</font></span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <font color="Maroon" title="c5">s</font></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">+</a> <span class="p4">(<span class="default"><font color="Maroon" title="c18">l</font> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <font color="Maroon" title="c6">t</font></span>)</span></span>)</span> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c7">o</font></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span>
<b>by </b><i><a class="txt" href="brouwer.html#E9:13"><i><font color="Green" title="E13">A7</font></i></a>, <a class="ref" href="topreal9.html#T9" title="TOPREAL9:th.9">TOPREAL9:9</a></i>
<br/>.= 
<span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default">1 <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <font color="Maroon" title="c5">s</font></span>)</span> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <span class="p5">(<span class="default"><font color="Maroon" title="c18">l</font> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <font color="Maroon" title="c5">s</font></span>)</span></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">+</a> <span class="p4">(<span class="default"><font color="Maroon" title="c18">l</font> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <font color="Maroon" title="c6">t</font></span>)</span></span>)</span> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c7">o</font></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span>
<b>by </b><i><a class="ref" href="euclid.html#T54" title="EUCLID:th.54">EUCLID:54</a></i>
<br/>.= 
<span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</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="c5">s</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <span class="p5">(<span class="default"><font color="Maroon" title="c18">l</font> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <font color="Maroon" title="c5">s</font></span>)</span></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">+</a> <span class="p4">(<span class="default"><font color="Maroon" title="c18">l</font> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <font color="Maroon" title="c6">t</font></span>)</span></span>)</span> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c7">o</font></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span>
<b>by </b><i><a class="ref" href="euclid.html#T33" title="EUCLID:th.33">EUCLID:33</a></i>
<br/>.= 
<span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c5">s</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c18">l</font> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <font color="Maroon" title="c5">s</font></span>)</span> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <span class="p5">(<span class="default"><font color="Maroon" title="c18">l</font> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <font color="Maroon" title="c6">t</font></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c7">o</font></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span>
<b>by </b><i><a class="ref" href="euclid.html#T51" title="EUCLID:th.51">EUCLID:51</a></i>
<br/>.= 
<span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c5">s</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c7">o</font></span>)</span> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c18">l</font> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <font color="Maroon" title="c5">s</font></span>)</span> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <span class="p4">(<span class="default"><font color="Maroon" title="c18">l</font> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <font color="Maroon" title="c6">t</font></span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span>
<b>by </b><i><a class="ref" href="topreal9.html#T1" title="TOPREAL9:th.1">TOPREAL9:1</a></i>
<br/>.= 
<span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c5">s</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c7">o</font></span>)</span> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c18">l</font> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c5">s</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c6">t</font></span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span>
<b>by </b><i><a class="ref" href="euclid.html#T53" title="EUCLID:th.53">EUCLID:53</a></i>
<br/>.= 
<span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c5">s</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c7">o</font></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">+</a> <span class="p3">(<span class="default"><a href="euclid.html#K19" title="EUCLID:func.19">-</a> <span class="p4">(<span class="default"><font color="Maroon" title="c18">l</font> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <span class="p5">(<span class="default"><font color="Maroon" title="c5">s</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c6">t</font></span>)</span></span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span>
<b>by </b><i><a class="ref" href="euclid.html#T45" title="EUCLID:th.45">EUCLID:45</a></i>
<br/>.= 
<span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c5">s</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c7">o</font></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c18">l</font> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <span class="p4">(<span class="default"><a href="euclid.html#K19" title="EUCLID:func.19">-</a> <span class="p5">(<span class="default"><font color="Maroon" title="c5">s</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c6">t</font></span>)</span></span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span>
<b>by </b><i><a class="ref" href="euclid.html#T44" title="EUCLID:th.44">EUCLID:44</a></i>
<br/>.= 
<span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c5">s</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c7">o</font></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c18">l</font> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c6">t</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c5">s</font></span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span>
<b>by </b><i><a class="ref" href="euclid.html#T48" title="EUCLID:th.48">EUCLID:48</a></i>
;<br/>
<b>then </b><font color="Maroon" title="c3">r</font> <a href="square_1.html#K3" title="SQUARE_1:func.3">^2</a>  = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c5">s</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c7">o</font></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p2">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p3"><a href="euclid_2.html#K2" title="EUCLID_2:func.2">|(</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c18">l</font> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <span class="p5">(<span class="default"><font color="Maroon" title="c6">t</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c5">s</font></span>)</span></span>)</span>,<span class="p4">(<span class="default"><font color="Maroon" title="c5">s</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c7">o</font></span>)</span></span><a href="euclid_2.html#K2" title="EUCLID_2:func.2">)|</a></span></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><span class="p2"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c18">l</font> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c6">t</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c5">s</font></span>)</span></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span>
<b>by </b><i><a class="ref" href="euclid_2.html#T67" title="EUCLID_2:th.67">EUCLID_2:67</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c5">s</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c7">o</font></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p2">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c18">l</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p4"><a href="euclid_2.html#K2" title="EUCLID_2:func.2">|(</a><span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c6">t</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c5">s</font></span>)</span>,<span class="p5">(<span class="default"><font color="Maroon" title="c5">s</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c7">o</font></span>)</span></span><a href="euclid_2.html#K2" title="EUCLID_2:func.2">)|</a></span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><span class="p2"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c18">l</font> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c6">t</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c5">s</font></span>)</span></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span>
<b>by </b><i><a class="ref" href="euclid_2.html#T41" title="EUCLID_2:th.41">EUCLID_2:41</a></i>
;<br/>
<a NAME="E34:13"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p4">(<span class="default"><a href="euclid.html#K11" title="EUCLID:func.11">sqr</a> <span class="p5">(<span class="default"><font color="Maroon" title="c9">T</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c8">S</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c18">l</font> <a href="square_1.html#K3" title="SQUARE_1:func.3">^2</a> </span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p4"><a href="euclid_2.html#K2" title="EUCLID_2:func.2">|(</a><span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c6">t</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c5">s</font></span>)</span>,<span class="p5">(<span class="default"><font color="Maroon" title="c5">s</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c7">o</font></span>)</span></span><a href="euclid_2.html#K2" title="EUCLID_2:func.2">)|</a></span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c18">l</font></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p3">(<span class="default"><a href="euclid.html#K11" title="EUCLID:func.11">sqr</a> <span class="p4">(<span class="default"><font color="Maroon" title="c8">S</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c10">O</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">r</font> <a href="square_1.html#K3" title="SQUARE_1:func.3">^2</a> </span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<b>by </b><i><a class="txt" href="brouwer.html#E1:13"><i><font color="Green" title="E7">A1</font></i></a>, <a class="txt" href="brouwer.html#E20:13"><i><font color="Green" title="E22">A16</font></i></a>, <a class="txt" href="brouwer.html#E22:13"><i><font color="Green" title="E23">A17</font></i></a>, <a class="txt" href="brouwer.html#E30:13"><i><font color="Green" title="E28">A22</font></i></a></i>;<br/>
<a NAME="E35:13"/><b>then </b><i><font color="Green" title="E29">A23</font></i>: 
 <a href="polyeq_1.html#K3" title="POLYEQ_1:func.3">Polynom</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="euclid.html#K11" title="EUCLID:func.11">sqr</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">T</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c8">S</font></span>)</span></span>)</span></span>)</span>,<span class="p1">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2"><a href="euclid_2.html#K2" title="EUCLID_2:func.2">|(</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c6">t</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c5">s</font></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon" title="c5">s</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c7">o</font></span>)</span></span><a href="euclid_2.html#K2" title="EUCLID_2:func.2">)|</a></span></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p3">(<span class="default"><a href="euclid.html#K11" title="EUCLID:func.11">sqr</a> <span class="p4">(<span class="default"><font color="Maroon" title="c8">S</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c10">O</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">r</font> <a href="square_1.html#K3" title="SQUARE_1:func.3">^2</a> </span>)</span></span>)</span>,<font color="Maroon" title="c18">l</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<b>by </b><i><a class="ref" href="polyeq_1.html#D2" title="POLYEQ_1:def.2">POLYEQ_1:def 2</a></i>;<br/>
<a NAME="E36:13"/>
<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="euclid.html#K11" title="EUCLID:func.11">sqr</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">T</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c8">S</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p3">(<span class="default"><a href="euclid.html#K11" title="EUCLID:func.11">sqr</a> <span class="p4">(<span class="default"><font color="Maroon" title="c8">S</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c10">O</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">r</font> <a href="square_1.html#K3" title="SQUARE_1:func.3">^2</a> </span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<b>by </b><i><a class="txt" href="brouwer.html#E13:13"><i><font color="Green" title="E17">A11</font></i></a>, <a class="txt" href="brouwer.html#E24:13"><i><font color="Green" title="E25">A19</font></i></a>, <a class="ref" href="xreal_1.html#T133" title="XREAL_1:th.133">XREAL_1:133</a></i>;<br/>
<a NAME="E37:13"/><b>then </b>
 <a href="real_1.html#K1" title="REAL_1:func.1">-</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p3">(<span class="default"><a href="euclid.html#K11" title="EUCLID:func.11">sqr</a> <span class="p4">(<span class="default"><font color="Maroon" title="c9">T</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c8">S</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p4">(<span class="default"><a href="euclid.html#K11" title="EUCLID:func.11">sqr</a> <span class="p5">(<span class="default"><font color="Maroon" title="c8">S</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c10">O</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">r</font> <a href="square_1.html#K3" title="SQUARE_1:func.3">^2</a> </span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="xreal_1.html#T26" title="XREAL_1:th.26">XREAL_1:26</a></i>;<br/>
<a NAME="E38:13"/><b>then </b>
<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2"><a href="euclid_2.html#K2" title="EUCLID_2:func.2">|(</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c6">t</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c5">s</font></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon" title="c5">s</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c7">o</font></span>)</span></span><a href="euclid_2.html#K2" title="EUCLID_2:func.2">)|</a></span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p4">(<span class="default"><a href="euclid.html#K11" title="EUCLID:func.11">sqr</a> <span class="p5">(<span class="default"><font color="Maroon" title="c9">T</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c8">S</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p5">(<span class="default"><a href="euclid.html#K11" title="EUCLID:func.11">sqr</a> <span class="p0">(<span class="default"><font color="Maroon" title="c8">S</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c10">O</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">r</font> <a href="square_1.html#K3" title="SQUARE_1:func.3">^2</a> </span>)</span></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="xreal_1.html#T9" title="XREAL_1:th.9">XREAL_1:9</a></i>;<br/>
<b>then </b><i><font color="Green" title="E30">A24</font></i>: <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p3">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p4"><a href="euclid_2.html#K2" title="EUCLID_2:func.2">|(</a><span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c6">t</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c5">s</font></span>)</span>,<span class="p5">(<span class="default"><font color="Maroon" title="c5">s</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c7">o</font></span>)</span></span><a href="euclid_2.html#K2" title="EUCLID_2:func.2">)|</a></span></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p2">(<span class="default"><a href="square_1.html#K7" title="SQUARE_1:func.7">sqrt</a> <span class="p3">(<span class="default"><a href="quin_1.html#K2" title="QUIN_1:func.2">delta</a> <span class="p4">(<span class="default"><a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p5">(<span class="default"><a href="euclid.html#K11" title="EUCLID:func.11">sqr</a> <span class="p0">(<span class="default"><font color="Maroon" title="c9">T</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c8">S</font></span>)</span></span>)</span></span>)</span>,<span class="p4">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p5"><a href="euclid_2.html#K2" title="EUCLID_2:func.2">|(</a><span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c6">t</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c5">s</font></span>)</span>,<span class="p0">(<span class="default"><font color="Maroon" title="c5">s</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c7">o</font></span>)</span></span><a href="euclid_2.html#K2" title="EUCLID_2:func.2">)|</a></span></span>)</span>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p0">(<span class="default"><a href="euclid.html#K11" title="EUCLID:func.11">sqr</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">S</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c10">O</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p5">(<span class="default"><font color="Maroon" title="c3">r</font> <a href="square_1.html#K3" title="SQUARE_1:func.3">^2</a> </span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</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="euclid.html#K11" title="EUCLID:func.11">sqr</a> <span class="p4">(<span class="default"><font color="Maroon" title="c9">T</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c8">S</font></span>)</span></span>)</span></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p3">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p4"><a href="euclid_2.html#K2" title="EUCLID_2:func.2">|(</a><span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c6">t</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c5">s</font></span>)</span>,<span class="p5">(<span class="default"><font color="Maroon" title="c5">s</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c7">o</font></span>)</span></span><a href="euclid_2.html#K2" title="EUCLID_2:func.2">)|</a></span></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="square_1.html#K7" title="SQUARE_1:func.7">sqrt</a> 4</span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p3">(<span class="default"><a href="square_1.html#K7" title="SQUARE_1:func.7">sqrt</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0"><a href="euclid_2.html#K2" title="EUCLID_2:func.2">|(</a><span class="default"><span class="p1">(<span class="default"><font color="Maroon" title="c6">t</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c5">s</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c5">s</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c7">o</font></span>)</span></span><a href="euclid_2.html#K2" title="EUCLID_2:func.2">)|</a></span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p1">(<span class="default"><a href="euclid.html#K11" title="EUCLID:func.11">sqr</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">T</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c8">S</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p0">(<span class="default"><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="euclid.html#K11" title="EUCLID:func.11">sqr</a> <span class="p3">(<span class="default"><font color="Maroon" title="c8">S</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c10">O</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">r</font> <a href="square_1.html#K3" title="SQUARE_1:func.3">^2</a> </span>)</span></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</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="euclid.html#K11" title="EUCLID:func.11">sqr</a> <span class="p4">(<span class="default"><font color="Maroon" title="c9">T</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c8">S</font></span>)</span></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="brouwer.html#E14:13"><i><font color="Green" title="E18">A12</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">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p3">(<span class="default"><span class="p4"><a href="euclid_2.html#K2" title="EUCLID_2:func.2">|(</a><span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c6">t</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c5">s</font></span>)</span>,<span class="p5">(<span class="default"><font color="Maroon" title="c5">s</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c7">o</font></span>)</span></span><a href="euclid_2.html#K2" title="EUCLID_2:func.2">)|</a></span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p4">(<span class="default"><a href="square_1.html#K7" title="SQUARE_1:func.7">sqrt</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><span class="p1"><a href="euclid_2.html#K2" title="EUCLID_2:func.2">|(</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">t</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c5">s</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c5">s</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c7">o</font></span>)</span></span><a href="euclid_2.html#K2" title="EUCLID_2:func.2">)|</a></span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p0">(<span class="default"><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="euclid.html#K11" title="EUCLID:func.11">sqr</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">T</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c8">S</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p3">(<span class="default"><a href="euclid.html#K11" title="EUCLID:func.11">sqr</a> <span class="p4">(<span class="default"><font color="Maroon" title="c8">S</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c10">O</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">r</font> <a href="square_1.html#K3" title="SQUARE_1:func.3">^2</a> </span>)</span></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</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="euclid.html#K11" title="EUCLID:func.11">sqr</a> <span class="p4">(<span class="default"><font color="Maroon" title="c9">T</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c8">S</font></span>)</span></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="square_1.html#T85" title="SQUARE_1:th.85">SQUARE_1:85</a></i>
<br/>.= 
<font color="Maroon" title="c2">a</font>
<b>by </b><i><a class="txt" href="brouwer.html#E5:13"><i><font color="Green" title="E11">A5</font></i></a>, <a class="ref" href="xcmplx_1.html#T92" title="XCMPLX_1:th.92">XCMPLX_1:92</a></i>
;<br/>
<a NAME="E40:13"/><i><font color="Green" title="E31">A25</font></i>: 
( <font color="Maroon" title="c18">l</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p3">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p4"><a href="euclid_2.html#K2" title="EUCLID_2:func.2">|(</a><span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c6">t</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c5">s</font></span>)</span>,<span class="p5">(<span class="default"><font color="Maroon" title="c5">s</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c7">o</font></span>)</span></span><a href="euclid_2.html#K2" title="EUCLID_2:func.2">)|</a></span></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default"><a href="square_1.html#K7" title="SQUARE_1:func.7">sqrt</a> <span class="p3">(<span class="default"><a href="quin_1.html#K2" title="QUIN_1:func.2">delta</a> <span class="p4">(<span class="default"><a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p5">(<span class="default"><a href="euclid.html#K11" title="EUCLID:func.11">sqr</a> <span class="p0">(<span class="default"><font color="Maroon" title="c9">T</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c8">S</font></span>)</span></span>)</span></span>)</span>,<span class="p4">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p5"><a href="euclid_2.html#K2" title="EUCLID_2:func.2">|(</a><span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c6">t</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c5">s</font></span>)</span>,<span class="p0">(<span class="default"><font color="Maroon" title="c5">s</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c7">o</font></span>)</span></span><a href="euclid_2.html#K2" title="EUCLID_2:func.2">)|</a></span></span>)</span>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p0">(<span class="default"><a href="euclid.html#K11" title="EUCLID:func.11">sqr</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">S</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c10">O</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p5">(<span class="default"><font color="Maroon" title="c3">r</font> <a href="square_1.html#K3" title="SQUARE_1:func.3">^2</a> </span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</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="euclid.html#K11" title="EUCLID:func.11">sqr</a> <span class="p4">(<span class="default"><font color="Maroon" title="c9">T</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c8">S</font></span>)</span></span>)</span></span>)</span></span>)</span> or <font color="Maroon" title="c18">l</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p3">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p4"><a href="euclid_2.html#K2" title="EUCLID_2:func.2">|(</a><span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c6">t</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c5">s</font></span>)</span>,<span class="p5">(<span class="default"><font color="Maroon" title="c5">s</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c7">o</font></span>)</span></span><a href="euclid_2.html#K2" title="EUCLID_2:func.2">)|</a></span></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p2">(<span class="default"><a href="square_1.html#K7" title="SQUARE_1:func.7">sqrt</a> <span class="p3">(<span class="default"><a href="quin_1.html#K2" title="QUIN_1:func.2">delta</a> <span class="p4">(<span class="default"><a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p5">(<span class="default"><a href="euclid.html#K11" title="EUCLID:func.11">sqr</a> <span class="p0">(<span class="default"><font color="Maroon" title="c9">T</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c8">S</font></span>)</span></span>)</span></span>)</span>,<span class="p4">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p5"><a href="euclid_2.html#K2" title="EUCLID_2:func.2">|(</a><span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c6">t</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c5">s</font></span>)</span>,<span class="p0">(<span class="default"><font color="Maroon" title="c5">s</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c7">o</font></span>)</span></span><a href="euclid_2.html#K2" title="EUCLID_2:func.2">)|</a></span></span>)</span>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p0">(<span class="default"><a href="euclid.html#K11" title="EUCLID:func.11">sqr</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">S</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c10">O</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p5">(<span class="default"><font color="Maroon" title="c3">r</font> <a href="square_1.html#K3" title="SQUARE_1:func.3">^2</a> </span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</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="euclid.html#K11" title="EUCLID:func.11">sqr</a> <span class="p4">(<span class="default"><font color="Maroon" title="c9">T</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c8">S</font></span>)</span></span>)</span></span>)</span></span>)</span> )
 
<b>by </b><i><a class="txt" href="brouwer.html#E13:13"><i><font color="Green" title="E17">A11</font></i></a>, <a class="txt" href="brouwer.html#E25:13"><i><font color="Green" title="E26">A20</font></i></a>, <a class="txt" href="brouwer.html#E28:13"><i><font color="Green" title="E27">A21</font></i></a>, <a class="txt" href="brouwer.html#E35:13"><i><font color="Green" title="E29">A23</font></i></a>, <a class="ref" href="polyeq_1.html#T5" title="POLYEQ_1:th.5">POLYEQ_1:5</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="13_8"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c19">x</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>thus </b> <a href="polyeq_1.html#K3" title="POLYEQ_1:func.3">Polynom</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="euclid.html#K11" title="EUCLID:func.11">sqr</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">T</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c8">S</font></span>)</span></span>)</span></span>)</span>,<span class="p1">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2"><a href="euclid_2.html#K2" title="EUCLID_2:func.2">|(</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c6">t</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c5">s</font></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon" title="c5">s</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c7">o</font></span>)</span></span><a href="euclid_2.html#K2" title="EUCLID_2:func.2">)|</a></span></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p3">(<span class="default"><a href="euclid.html#K11" title="EUCLID:func.11">sqr</a> <span class="p4">(<span class="default"><font color="Maroon" title="c8">S</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c10">O</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">r</font> <a href="square_1.html#K3" title="SQUARE_1:func.3">^2</a> </span>)</span></span>)</span>,<font color="Maroon" title="c19">x</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p4">(<span class="default"><a href="euclid.html#K11" title="EUCLID:func.11">sqr</a> <span class="p5">(<span class="default"><font color="Maroon" title="c9">T</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c8">S</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c19">x</font> <a href="square_1.html#K3" title="SQUARE_1:func.3">^2</a> </span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p4"><a href="euclid_2.html#K2" title="EUCLID_2:func.2">|(</a><span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c6">t</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c5">s</font></span>)</span>,<span class="p5">(<span class="default"><font color="Maroon" title="c5">s</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c7">o</font></span>)</span></span><a href="euclid_2.html#K2" title="EUCLID_2:func.2">)|</a></span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c19">x</font></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p3">(<span class="default"><a href="euclid.html#K11" title="EUCLID:func.11">sqr</a> <span class="p4">(<span class="default"><font color="Maroon" title="c8">S</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c10">O</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">r</font> <a href="square_1.html#K3" title="SQUARE_1:func.3">^2</a> </span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="polyeq_1.html#D2" title="POLYEQ_1:def.2">POLYEQ_1:def 2</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p3">(<span class="default"><a href="euclid.html#K11" title="EUCLID:func.11">sqr</a> <span class="p4">(<span class="default"><font color="Maroon" title="c9">T</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c8">S</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c19">x</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p0">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1"><a href="euclid_2.html#K2" title="EUCLID_2:func.2">|(</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">t</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c5">s</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c5">s</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c7">o</font></span>)</span></span><a href="euclid_2.html#K2" title="EUCLID_2:func.2">)|</a></span></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p5">(<span class="default"><a href="square_1.html#K7" title="SQUARE_1:func.7">sqrt</a> <span class="p0">(<span class="default"><a href="quin_1.html#K2" title="QUIN_1:func.2">delta</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="euclid.html#K11" title="EUCLID:func.11">sqr</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">T</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c8">S</font></span>)</span></span>)</span></span>)</span>,<span class="p1">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2"><a href="euclid_2.html#K2" title="EUCLID_2:func.2">|(</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c6">t</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c5">s</font></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon" title="c5">s</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c7">o</font></span>)</span></span><a href="euclid_2.html#K2" title="EUCLID_2:func.2">)|</a></span></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p3">(<span class="default"><a href="euclid.html#K11" title="EUCLID:func.11">sqr</a> <span class="p4">(<span class="default"><font color="Maroon" title="c8">S</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c10">O</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">r</font> <a href="square_1.html#K3" title="SQUARE_1:func.3">^2</a> </span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p4">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p5">(<span class="default"><a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p0">(<span class="default"><a href="euclid.html#K11" title="EUCLID:func.11">sqr</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">T</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c8">S</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c19">x</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p5">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p0"><a href="euclid_2.html#K2" title="EUCLID_2:func.2">|(</a><span class="default"><span class="p1">(<span class="default"><font color="Maroon" title="c6">t</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c5">s</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c5">s</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c7">o</font></span>)</span></span><a href="euclid_2.html#K2" title="EUCLID_2:func.2">)|</a></span></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p4">(<span class="default"><a href="square_1.html#K7" title="SQUARE_1:func.7">sqrt</a> <span class="p5">(<span class="default"><a href="quin_1.html#K2" title="QUIN_1:func.2">delta</a> <span class="p0">(<span class="default"><a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p1">(<span class="default"><a href="euclid.html#K11" title="EUCLID:func.11">sqr</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">T</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c8">S</font></span>)</span></span>)</span></span>)</span>,<span class="p0">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1"><a href="euclid_2.html#K2" title="EUCLID_2:func.2">|(</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">t</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c5">s</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c5">s</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c7">o</font></span>)</span></span><a href="euclid_2.html#K2" title="EUCLID_2:func.2">)|</a></span></span>)</span>,<span class="p0">(<span class="default"><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="euclid.html#K11" title="EUCLID:func.11">sqr</a> <span class="p3">(<span class="default"><font color="Maroon" title="c8">S</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c10">O</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">r</font> <a href="square_1.html#K3" title="SQUARE_1:func.3">^2</a> </span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p3">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p4">(<span class="default"><a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p5">(<span class="default"><a href="euclid.html#K11" title="EUCLID:func.11">sqr</a> <span class="p0">(<span class="default"><font color="Maroon" title="c9">T</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c8">S</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="brouwer.html#E13:13"><i><font color="Green" title="E17">A11</font></i></a>, <a class="txt" href="brouwer.html#E25:13"><i><font color="Green" title="E26">A20</font></i></a>, <a class="txt" href="brouwer.html#E28:13"><i><font color="Green" title="E27">A21</font></i></a>, <a class="ref" href="quin_1.html#T16" title="QUIN_1:th.16">QUIN_1:16</a></i>
<br/>.= 
<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="euclid.html#K11" title="EUCLID:func.11">sqr</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">T</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c8">S</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c19">x</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p0">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1"><a href="euclid_2.html#K2" title="EUCLID_2:func.2">|(</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">t</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c5">s</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c5">s</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c7">o</font></span>)</span></span><a href="euclid_2.html#K2" title="EUCLID_2:func.2">)|</a></span></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p5">(<span class="default"><a href="square_1.html#K7" title="SQUARE_1:func.7">sqrt</a> <span class="p0">(<span class="default"><a href="quin_1.html#K2" title="QUIN_1:func.2">delta</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="euclid.html#K11" title="EUCLID:func.11">sqr</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">T</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c8">S</font></span>)</span></span>)</span></span>)</span>,<span class="p1">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2"><a href="euclid_2.html#K2" title="EUCLID_2:func.2">|(</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c6">t</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c5">s</font></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon" title="c5">s</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c7">o</font></span>)</span></span><a href="euclid_2.html#K2" title="EUCLID_2:func.2">)|</a></span></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p3">(<span class="default"><a href="euclid.html#K11" title="EUCLID:func.11">sqr</a> <span class="p4">(<span class="default"><font color="Maroon" title="c8">S</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c10">O</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">r</font> <a href="square_1.html#K3" title="SQUARE_1:func.3">^2</a> </span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p4">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p5">(<span class="default"><a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p0">(<span class="default"><a href="euclid.html#K11" title="EUCLID:func.11">sqr</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">T</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c8">S</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c19">x</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p0">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1"><a href="euclid_2.html#K2" title="EUCLID_2:func.2">|(</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">t</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c5">s</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c5">s</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c7">o</font></span>)</span></span><a href="euclid_2.html#K2" title="EUCLID_2:func.2">)|</a></span></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p5">(<span class="default"><a href="square_1.html#K7" title="SQUARE_1:func.7">sqrt</a> <span class="p0">(<span class="default"><a href="quin_1.html#K2" title="QUIN_1:func.2">delta</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="euclid.html#K11" title="EUCLID:func.11">sqr</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">T</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c8">S</font></span>)</span></span>)</span></span>)</span>,<span class="p1">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2"><a href="euclid_2.html#K2" title="EUCLID_2:func.2">|(</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c6">t</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c5">s</font></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon" title="c5">s</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c7">o</font></span>)</span></span><a href="euclid_2.html#K2" title="EUCLID_2:func.2">)|</a></span></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p3">(<span class="default"><a href="euclid.html#K11" title="EUCLID:func.11">sqr</a> <span class="p4">(<span class="default"><font color="Maroon" title="c8">S</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c10">O</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">r</font> <a href="square_1.html#K3" title="SQUARE_1:func.3">^2</a> </span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p4">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p5">(<span class="default"><a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p0">(<span class="default"><a href="euclid.html#K11" title="EUCLID:func.11">sqr</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">T</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c8">S</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span>

<br/>.= 
 <a href="polyeq_1.html#K5" title="POLYEQ_1:func.5">Quard</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="euclid.html#K11" title="EUCLID:func.11">sqr</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">T</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c8">S</font></span>)</span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p4">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p5"><a href="euclid_2.html#K2" title="EUCLID_2:func.2">|(</a><span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c6">t</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c5">s</font></span>)</span>,<span class="p0">(<span class="default"><font color="Maroon" title="c5">s</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c7">o</font></span>)</span></span><a href="euclid_2.html#K2" title="EUCLID_2:func.2">)|</a></span></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p3">(<span class="default"><a href="square_1.html#K7" title="SQUARE_1:func.7">sqrt</a> <span class="p4">(<span class="default"><a href="quin_1.html#K2" title="QUIN_1:func.2">delta</a> <span class="p5">(<span class="default"><a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p0">(<span class="default"><a href="euclid.html#K11" title="EUCLID:func.11">sqr</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">T</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c8">S</font></span>)</span></span>)</span></span>)</span>,<span class="p5">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p0"><a href="euclid_2.html#K2" title="EUCLID_2:func.2">|(</a><span class="default"><span class="p1">(<span class="default"><font color="Maroon" title="c6">t</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c5">s</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c5">s</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c7">o</font></span>)</span></span><a href="euclid_2.html#K2" title="EUCLID_2:func.2">)|</a></span></span>)</span>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p1">(<span class="default"><a href="euclid.html#K11" title="EUCLID:func.11">sqr</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">S</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c10">O</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p0">(<span class="default"><font color="Maroon" title="c3">r</font> <a href="square_1.html#K3" title="SQUARE_1:func.3">^2</a> </span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p2">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p3">(<span class="default"><a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p4">(<span class="default"><a href="euclid.html#K11" title="EUCLID:func.11">sqr</a> <span class="p5">(<span class="default"><font color="Maroon" title="c9">T</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c8">S</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p4">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p5"><a href="euclid_2.html#K2" title="EUCLID_2:func.2">|(</a><span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c6">t</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c5">s</font></span>)</span>,<span class="p0">(<span class="default"><font color="Maroon" title="c5">s</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c7">o</font></span>)</span></span><a href="euclid_2.html#K2" title="EUCLID_2:func.2">)|</a></span></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p3">(<span class="default"><a href="square_1.html#K7" title="SQUARE_1:func.7">sqrt</a> <span class="p4">(<span class="default"><a href="quin_1.html#K2" title="QUIN_1:func.2">delta</a> <span class="p5">(<span class="default"><a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p0">(<span class="default"><a href="euclid.html#K11" title="EUCLID:func.11">sqr</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">T</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c8">S</font></span>)</span></span>)</span></span>)</span>,<span class="p5">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p0"><a href="euclid_2.html#K2" title="EUCLID_2:func.2">|(</a><span class="default"><span class="p1">(<span class="default"><font color="Maroon" title="c6">t</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c5">s</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c5">s</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c7">o</font></span>)</span></span><a href="euclid_2.html#K2" title="EUCLID_2:func.2">)|</a></span></span>)</span>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p1">(<span class="default"><a href="euclid.html#K11" title="EUCLID:func.11">sqr</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">S</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c10">O</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p0">(<span class="default"><font color="Maroon" title="c3">r</font> <a href="square_1.html#K3" title="SQUARE_1:func.3">^2</a> </span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p2">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p3">(<span class="default"><a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p4">(<span class="default"><a href="euclid.html#K11" title="EUCLID:func.11">sqr</a> <span class="p5">(<span class="default"><font color="Maroon" title="c9">T</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c8">S</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span>,<font color="Maroon" title="c19">x</font>
<b>by </b><i><a class="ref" href="polyeq_1.html#D3" title="POLYEQ_1:def.3">POLYEQ_1:def 3</a></i>
;<br/></div><b>end;</b></div>
<a NAME="E42:13"/><b>then </b><i><font color="Green" title="E32">A26</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p3">(<span class="default"><a href="euclid.html#K11" title="EUCLID:func.11">sqr</a> <span class="p4">(<span class="default"><font color="Maroon" title="c8">S</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c10">O</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">r</font> <a href="square_1.html#K3" title="SQUARE_1:func.3">^2</a> </span>)</span></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</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="euclid.html#K11" title="EUCLID:func.11">sqr</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">T</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c8">S</font></span>)</span></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"><span class="p3">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p4">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p5"><a href="euclid_2.html#K2" title="EUCLID_2:func.2">|(</a><span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c6">t</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c5">s</font></span>)</span>,<span class="p0">(<span class="default"><font color="Maroon" title="c5">s</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c7">o</font></span>)</span></span><a href="euclid_2.html#K2" title="EUCLID_2:func.2">)|</a></span></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p3">(<span class="default"><a href="square_1.html#K7" title="SQUARE_1:func.7">sqrt</a> <span class="p4">(<span class="default"><a href="quin_1.html#K2" title="QUIN_1:func.2">delta</a> <span class="p5">(<span class="default"><a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p0">(<span class="default"><a href="euclid.html#K11" title="EUCLID:func.11">sqr</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">T</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c8">S</font></span>)</span></span>)</span></span>)</span>,<span class="p5">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p0"><a href="euclid_2.html#K2" title="EUCLID_2:func.2">|(</a><span class="default"><span class="p1">(<span class="default"><font color="Maroon" title="c6">t</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c5">s</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c5">s</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c7">o</font></span>)</span></span><a href="euclid_2.html#K2" title="EUCLID_2:func.2">)|</a></span></span>)</span>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p1">(<span class="default"><a href="euclid.html#K11" title="EUCLID:func.11">sqr</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">S</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c10">O</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p0">(<span class="default"><font color="Maroon" title="c3">r</font> <a href="square_1.html#K3" title="SQUARE_1:func.3">^2</a> </span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p2">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p3">(<span class="default"><a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p4">(<span class="default"><a href="euclid.html#K11" title="EUCLID:func.11">sqr</a> <span class="p5">(<span class="default"><font color="Maroon" title="c9">T</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c8">S</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p4">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p5"><a href="euclid_2.html#K2" title="EUCLID_2:func.2">|(</a><span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c6">t</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c5">s</font></span>)</span>,<span class="p0">(<span class="default"><font color="Maroon" title="c5">s</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c7">o</font></span>)</span></span><a href="euclid_2.html#K2" title="EUCLID_2:func.2">)|</a></span></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p3">(<span class="default"><a href="square_1.html#K7" title="SQUARE_1:func.7">sqrt</a> <span class="p4">(<span class="default"><a href="quin_1.html#K2" title="QUIN_1:func.2">delta</a> <span class="p5">(<span class="default"><a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p0">(<span class="default"><a href="euclid.html#K11" title="EUCLID:func.11">sqr</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">T</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c8">S</font></span>)</span></span>)</span></span>)</span>,<span class="p5">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p0"><a href="euclid_2.html#K2" title="EUCLID_2:func.2">|(</a><span class="default"><span class="p1">(<span class="default"><font color="Maroon" title="c6">t</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c5">s</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c5">s</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c7">o</font></span>)</span></span><a href="euclid_2.html#K2" title="EUCLID_2:func.2">)|</a></span></span>)</span>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p1">(<span class="default"><a href="euclid.html#K11" title="EUCLID:func.11">sqr</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">S</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c10">O</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p0">(<span class="default"><font color="Maroon" title="c3">r</font> <a href="square_1.html#K3" title="SQUARE_1:func.3">^2</a> </span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p2">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p3">(<span class="default"><a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p4">(<span class="default"><a href="euclid.html#K11" title="EUCLID:func.11">sqr</a> <span class="p5">(<span class="default"><font color="Maroon" title="c9">T</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c8">S</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="brouwer.html#E13:13"><i><font color="Green" title="E17">A11</font></i></a>, <a class="ref" href="polyeq_1.html#T11" title="POLYEQ_1:th.11">POLYEQ_1:11</a></i>;<br/>
<a NAME="E43:13"/><i><font color="Green" title="E33">A27</font></i>: 
2 <a href="real_1.html#K8" title="REAL_1:func.8">*</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="euclid.html#K11" title="EUCLID:func.11">sqr</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">T</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c8">S</font></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<b>by </b><i><a class="txt" href="brouwer.html#E13:13"><i><font color="Green" title="E17">A11</font></i></a>, <a class="ref" href="xreal_1.html#T131" title="XREAL_1:th.131">XREAL_1:131</a></i>;<br/>
<div><i><font color="Green" title="E34">A28</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="13_9"><b>now </b></a><div class="add"><b>set </b><font color="Maroon" title="c19">D</font> =  <a href="square_1.html#K7" title="SQUARE_1:func.7">sqrt</a> <span class="p1">(<span class="default"><a href="quin_1.html#K2" title="QUIN_1:func.2">delta</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="euclid.html#K11" title="EUCLID:func.11">sqr</a> <span class="p4">(<span class="default"><font color="Maroon" title="c9">T</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c8">S</font></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p3"><a href="euclid_2.html#K2" title="EUCLID_2:func.2">|(</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c6">t</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c5">s</font></span>)</span>,<span class="p4">(<span class="default"><font color="Maroon" title="c5">s</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c7">o</font></span>)</span></span><a href="euclid_2.html#K2" title="EUCLID_2:func.2">)|</a></span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p4">(<span class="default"><a href="euclid.html#K11" title="EUCLID:func.11">sqr</a> <span class="p5">(<span class="default"><font color="Maroon" title="c8">S</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c10">O</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">r</font> <a href="square_1.html#K3" title="SQUARE_1:func.3">^2</a> </span>)</span></span>)</span></span>)</span>;<br/><b>assume </b><a NAME="E1:13_9"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p3">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p4"><a href="euclid_2.html#K2" title="EUCLID_2:func.2">|(</a><span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c6">t</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c5">s</font></span>)</span>,<span class="p5">(<span class="default"><font color="Maroon" title="c5">s</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c7">o</font></span>)</span></span><a href="euclid_2.html#K2" title="EUCLID_2:func.2">)|</a></span></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default"><a href="square_1.html#K7" title="SQUARE_1:func.7">sqrt</a> <span class="p3">(<span class="default"><a href="quin_1.html#K2" title="QUIN_1:func.2">delta</a> <span class="p4">(<span class="default"><a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p5">(<span class="default"><a href="euclid.html#K11" title="EUCLID:func.11">sqr</a> <span class="p0">(<span class="default"><font color="Maroon" title="c9">T</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c8">S</font></span>)</span></span>)</span></span>)</span>,<span class="p4">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p5"><a href="euclid_2.html#K2" title="EUCLID_2:func.2">|(</a><span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c6">t</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c5">s</font></span>)</span>,<span class="p0">(<span class="default"><font color="Maroon" title="c5">s</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c7">o</font></span>)</span></span><a href="euclid_2.html#K2" title="EUCLID_2:func.2">)|</a></span></span>)</span>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p0">(<span class="default"><a href="euclid.html#K11" title="EUCLID:func.11">sqr</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">S</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c10">O</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p5">(<span class="default"><font color="Maroon" title="c3">r</font> <a href="square_1.html#K3" title="SQUARE_1:func.3">^2</a> </span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</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="euclid.html#K11" title="EUCLID:func.11">sqr</a> <span class="p4">(<span class="default"><font color="Maroon" title="c9">T</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c8">S</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p3">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p4"><a href="euclid_2.html#K2" title="EUCLID_2:func.2">|(</a><span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c6">t</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c5">s</font></span>)</span>,<span class="p5">(<span class="default"><font color="Maroon" title="c5">s</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c7">o</font></span>)</span></span><a href="euclid_2.html#K2" title="EUCLID_2:func.2">)|</a></span></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p2">(<span class="default"><a href="square_1.html#K7" title="SQUARE_1:func.7">sqrt</a> <span class="p3">(<span class="default"><a href="quin_1.html#K2" title="QUIN_1:func.2">delta</a> <span class="p4">(<span class="default"><a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p5">(<span class="default"><a href="euclid.html#K11" title="EUCLID:func.11">sqr</a> <span class="p0">(<span class="default"><font color="Maroon" title="c9">T</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c8">S</font></span>)</span></span>)</span></span>)</span>,<span class="p4">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p5"><a href="euclid_2.html#K2" title="EUCLID_2:func.2">|(</a><span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c6">t</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c5">s</font></span>)</span>,<span class="p0">(<span class="default"><font color="Maroon" title="c5">s</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c7">o</font></span>)</span></span><a href="euclid_2.html#K2" title="EUCLID_2:func.2">)|</a></span></span>)</span>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p0">(<span class="default"><a href="euclid.html#K11" title="EUCLID:func.11">sqr</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">S</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c10">O</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p5">(<span class="default"><font color="Maroon" title="c3">r</font> <a href="square_1.html#K3" title="SQUARE_1:func.3">^2</a> </span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</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="euclid.html#K11" title="EUCLID:func.11">sqr</a> <span class="p4">(<span class="default"><font color="Maroon" title="c9">T</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c8">S</font></span>)</span></span>)</span></span>)</span></span>)</span>
 ;<br/><a NAME="E2:13_9"/><b>then </b>
<span class="p1">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p2">(<span class="default"><a href="square_1.html#K7" title="SQUARE_1:func.7">sqrt</a> <span class="p3">(<span class="default"><a href="quin_1.html#K2" title="QUIN_1:func.2">delta</a> <span class="p4">(<span class="default"><a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p5">(<span class="default"><a href="euclid.html#K11" title="EUCLID:func.11">sqr</a> <span class="p0">(<span class="default"><font color="Maroon" title="c9">T</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c8">S</font></span>)</span></span>)</span></span>)</span>,<span class="p4">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p5"><a href="euclid_2.html#K2" title="EUCLID_2:func.2">|(</a><span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c6">t</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c5">s</font></span>)</span>,<span class="p0">(<span class="default"><font color="Maroon" title="c5">s</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c7">o</font></span>)</span></span><a href="euclid_2.html#K2" title="EUCLID_2:func.2">)|</a></span></span>)</span>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p0">(<span class="default"><a href="euclid.html#K11" title="EUCLID:func.11">sqr</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">S</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c10">O</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p5">(<span class="default"><font color="Maroon" title="c3">r</font> <a href="square_1.html#K3" title="SQUARE_1:func.3">^2</a> </span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2"><a href="euclid_2.html#K2" title="EUCLID_2:func.2">|(</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c6">t</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c5">s</font></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon" title="c5">s</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c7">o</font></span>)</span></span><a href="euclid_2.html#K2" title="EUCLID_2:func.2">)|</a></span></span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</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="quin_1.html#K2" title="QUIN_1:func.2">delta</a> <span class="p3">(<span class="default"><a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p4">(<span class="default"><a href="euclid.html#K11" title="EUCLID:func.11">sqr</a> <span class="p5">(<span class="default"><font color="Maroon" title="c9">T</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c8">S</font></span>)</span></span>)</span></span>)</span>,<span class="p3">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p4"><a href="euclid_2.html#K2" title="EUCLID_2:func.2">|(</a><span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c6">t</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c5">s</font></span>)</span>,<span class="p5">(<span class="default"><font color="Maroon" title="c5">s</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c7">o</font></span>)</span></span><a href="euclid_2.html#K2" title="EUCLID_2:func.2">)|</a></span></span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p5">(<span class="default"><a href="euclid.html#K11" title="EUCLID:func.11">sqr</a> <span class="p0">(<span class="default"><font color="Maroon" title="c8">S</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c10">O</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">r</font> <a href="square_1.html#K3" title="SQUARE_1:func.3">^2</a> </span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2"><a href="euclid_2.html#K2" title="EUCLID_2:func.2">|(</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c6">t</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c5">s</font></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon" title="c5">s</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c7">o</font></span>)</span></span><a href="euclid_2.html#K2" title="EUCLID_2:func.2">)|</a></span></span>)</span>
 <b>by </b><i><a class="txt" href="brouwer.html#E43:13"><i><font color="Green" title="E33">A27</font></i></a>, <a class="ref" href="real_1.html#T73" title="REAL_1:th.73">REAL_1:73</a></i>;<br/><a NAME="E3:13_9"/><b>then </b>
 <a href="real_1.html#K1" title="REAL_1:func.1">-</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="quin_1.html#K2" title="QUIN_1:func.2">delta</a> <span class="p3">(<span class="default"><a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p4">(<span class="default"><a href="euclid.html#K11" title="EUCLID:func.11">sqr</a> <span class="p5">(<span class="default"><font color="Maroon" title="c9">T</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c8">S</font></span>)</span></span>)</span></span>)</span>,<span class="p3">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p4"><a href="euclid_2.html#K2" title="EUCLID_2:func.2">|(</a><span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c6">t</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c5">s</font></span>)</span>,<span class="p5">(<span class="default"><font color="Maroon" title="c5">s</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c7">o</font></span>)</span></span><a href="euclid_2.html#K2" title="EUCLID_2:func.2">)|</a></span></span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p5">(<span class="default"><a href="euclid.html#K11" title="EUCLID:func.11">sqr</a> <span class="p0">(<span class="default"><font color="Maroon" title="c8">S</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c10">O</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">r</font> <a href="square_1.html#K3" title="SQUARE_1:func.3">^2</a> </span>)</span></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="square_1.html#K7" title="SQUARE_1:func.7">sqrt</a> <span class="p1">(<span class="default"><a href="quin_1.html#K2" title="QUIN_1:func.2">delta</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="euclid.html#K11" title="EUCLID:func.11">sqr</a> <span class="p4">(<span class="default"><font color="Maroon" title="c9">T</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c8">S</font></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p3"><a href="euclid_2.html#K2" title="EUCLID_2:func.2">|(</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c6">t</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c5">s</font></span>)</span>,<span class="p4">(<span class="default"><font color="Maroon" title="c5">s</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c7">o</font></span>)</span></span><a href="euclid_2.html#K2" title="EUCLID_2:func.2">)|</a></span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p4">(<span class="default"><a href="euclid.html#K11" title="EUCLID:func.11">sqr</a> <span class="p5">(<span class="default"><font color="Maroon" title="c8">S</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c10">O</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">r</font> <a href="square_1.html#K3" title="SQUARE_1:func.3">^2</a> </span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="xreal_1.html#T11" title="XREAL_1:th.11">XREAL_1:11</a></i>;<br/><a NAME="E4:13_9"/><b>then </b><i><font color="Green" title="E35">A29</font></i>: 
<span class="p1">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p2">(<span class="default"><a href="square_1.html#K7" title="SQUARE_1:func.7">sqrt</a> <span class="p3">(<span class="default"><a href="quin_1.html#K2" title="QUIN_1:func.2">delta</a> <span class="p4">(<span class="default"><a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p5">(<span class="default"><a href="euclid.html#K11" title="EUCLID:func.11">sqr</a> <span class="p0">(<span class="default"><font color="Maroon" title="c9">T</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c8">S</font></span>)</span></span>)</span></span>)</span>,<span class="p4">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p5"><a href="euclid_2.html#K2" title="EUCLID_2:func.2">|(</a><span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c6">t</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c5">s</font></span>)</span>,<span class="p0">(<span class="default"><font color="Maroon" title="c5">s</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c7">o</font></span>)</span></span><a href="euclid_2.html#K2" title="EUCLID_2:func.2">)|</a></span></span>)</span>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p0">(<span class="default"><a href="euclid.html#K11" title="EUCLID:func.11">sqr</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">S</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c10">O</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p5">(<span class="default"><font color="Maroon" title="c3">r</font> <a href="square_1.html#K3" title="SQUARE_1:func.3">^2</a> </span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</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="quin_1.html#K2" title="QUIN_1:func.2">delta</a> <span class="p3">(<span class="default"><a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p4">(<span class="default"><a href="euclid.html#K11" title="EUCLID:func.11">sqr</a> <span class="p5">(<span class="default"><font color="Maroon" title="c9">T</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c8">S</font></span>)</span></span>)</span></span>)</span>,<span class="p3">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p4"><a href="euclid_2.html#K2" title="EUCLID_2:func.2">|(</a><span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c6">t</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c5">s</font></span>)</span>,<span class="p5">(<span class="default"><font color="Maroon" title="c5">s</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c7">o</font></span>)</span></span><a href="euclid_2.html#K2" title="EUCLID_2:func.2">)|</a></span></span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p5">(<span class="default"><a href="euclid.html#K11" title="EUCLID:func.11">sqr</a> <span class="p0">(<span class="default"><font color="Maroon" title="c8">S</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c10">O</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">r</font> <a href="square_1.html#K3" title="SQUARE_1:func.3">^2</a> </span>)</span></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</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="quin_1.html#K2" title="QUIN_1:func.2">delta</a> <span class="p3">(<span class="default"><a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p4">(<span class="default"><a href="euclid.html#K11" title="EUCLID:func.11">sqr</a> <span class="p5">(<span class="default"><font color="Maroon" title="c9">T</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c8">S</font></span>)</span></span>)</span></span>)</span>,<span class="p3">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p4"><a href="euclid_2.html#K2" title="EUCLID_2:func.2">|(</a><span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c6">t</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c5">s</font></span>)</span>,<span class="p5">(<span class="default"><font color="Maroon" title="c5">s</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c7">o</font></span>)</span></span><a href="euclid_2.html#K2" title="EUCLID_2:func.2">)|</a></span></span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p5">(<span class="default"><a href="euclid.html#K11" title="EUCLID:func.11">sqr</a> <span class="p0">(<span class="default"><font color="Maroon" title="c8">S</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c10">O</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">r</font> <a href="square_1.html#K3" title="SQUARE_1:func.3">^2</a> </span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</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="quin_1.html#K2" title="QUIN_1:func.2">delta</a> <span class="p3">(<span class="default"><a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p4">(<span class="default"><a href="euclid.html#K11" title="EUCLID:func.11">sqr</a> <span class="p5">(<span class="default"><font color="Maroon" title="c9">T</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c8">S</font></span>)</span></span>)</span></span>)</span>,<span class="p3">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p4"><a href="euclid_2.html#K2" title="EUCLID_2:func.2">|(</a><span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c6">t</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c5">s</font></span>)</span>,<span class="p5">(<span class="default"><font color="Maroon" title="c5">s</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c7">o</font></span>)</span></span><a href="euclid_2.html#K2" title="EUCLID_2:func.2">)|</a></span></span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p5">(<span class="default"><a href="euclid.html#K11" title="EUCLID:func.11">sqr</a> <span class="p0">(<span class="default"><font color="Maroon" title="c8">S</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c10">O</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">r</font> <a href="square_1.html#K3" title="SQUARE_1:func.3">^2</a> </span>)</span></span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></i>;<br/><a NAME="E5:13_9"/>
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="square_1.html#K7" title="SQUARE_1:func.7">sqrt</a> <span class="p1">(<span class="default"><a href="quin_1.html#K2" title="QUIN_1:func.2">delta</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="euclid.html#K11" title="EUCLID:func.11">sqr</a> <span class="p4">(<span class="default"><font color="Maroon" title="c9">T</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c8">S</font></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p3"><a href="euclid_2.html#K2" title="EUCLID_2:func.2">|(</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c6">t</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c5">s</font></span>)</span>,<span class="p4">(<span class="default"><font color="Maroon" title="c5">s</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c7">o</font></span>)</span></span><a href="euclid_2.html#K2" title="EUCLID_2:func.2">)|</a></span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p4">(<span class="default"><a href="euclid.html#K11" title="EUCLID:func.11">sqr</a> <span class="p5">(<span class="default"><font color="Maroon" title="c8">S</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c10">O</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">r</font> <a href="square_1.html#K3" title="SQUARE_1:func.3">^2</a> </span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="brouwer.html#E25:13"><i><font color="Green" title="E26">A20</font></i></a>, <a class="txt" href="brouwer.html#E28:13"><i><font color="Green" title="E27">A21</font></i></a>, <a class="ref" href="square_1.html#D4" title="SQUARE_1:def.4">SQUARE_1:def 4</a></i>;<br/><a NAME="E6:13_9"/><b>then </b>
<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</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="quin_1.html#K2" title="QUIN_1:func.2">delta</a> <span class="p3">(<span class="default"><a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p4">(<span class="default"><a href="euclid.html#K11" title="EUCLID:func.11">sqr</a> <span class="p5">(<span class="default"><font color="Maroon" title="c9">T</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c8">S</font></span>)</span></span>)</span></span>)</span>,<span class="p3">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p4"><a href="euclid_2.html#K2" title="EUCLID_2:func.2">|(</a><span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c6">t</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c5">s</font></span>)</span>,<span class="p5">(<span class="default"><font color="Maroon" title="c5">s</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c7">o</font></span>)</span></span><a href="euclid_2.html#K2" title="EUCLID_2:func.2">)|</a></span></span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p5">(<span class="default"><a href="euclid.html#K11" title="EUCLID:func.11">sqr</a> <span class="p0">(<span class="default"><font color="Maroon" title="c8">S</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c10">O</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">r</font> <a href="square_1.html#K3" title="SQUARE_1:func.3">^2</a> </span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</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="quin_1.html#K2" title="QUIN_1:func.2">delta</a> <span class="p3">(<span class="default"><a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p4">(<span class="default"><a href="euclid.html#K11" title="EUCLID:func.11">sqr</a> <span class="p5">(<span class="default"><font color="Maroon" title="c9">T</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c8">S</font></span>)</span></span>)</span></span>)</span>,<span class="p3">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p4"><a href="euclid_2.html#K2" title="EUCLID_2:func.2">|(</a><span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c6">t</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c5">s</font></span>)</span>,<span class="p5">(<span class="default"><font color="Maroon" title="c5">s</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c7">o</font></span>)</span></span><a href="euclid_2.html#K2" title="EUCLID_2:func.2">)|</a></span></span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p5">(<span class="default"><a href="euclid.html#K11" title="EUCLID:func.11">sqr</a> <span class="p0">(<span class="default"><font color="Maroon" title="c8">S</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c10">O</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">r</font> <a href="square_1.html#K3" title="SQUARE_1:func.3">^2</a> </span>)</span></span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="xreal_1.html#T9" title="XREAL_1:th.9">XREAL_1:9</a></i>;<br/><b>hence </b><a NAME="E7:13_9"/>
contradiction
 <b>by </b><i><a class="txt" href="brouwer.html#E4:13_9"><i><font color="Green" title="E35">A29</font></i></a></i>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="13_10"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:13_10"/>
( <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="euclid.html#K11" title="EUCLID:func.11">sqr</a> <span class="p3">(<span class="default"><font color="Maroon" title="c8">S</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c10">O</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">r</font> <a href="square_1.html#K3" title="SQUARE_1:func.3">^2</a> </span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  or <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p3">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p4"><a href="euclid_2.html#K2" title="EUCLID_2:func.2">|(</a><span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c6">t</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c5">s</font></span>)</span>,<span class="p5">(<span class="default"><font color="Maroon" title="c5">s</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c7">o</font></span>)</span></span><a href="euclid_2.html#K2" title="EUCLID_2:func.2">)|</a></span></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default"><a href="square_1.html#K7" title="SQUARE_1:func.7">sqrt</a> <span class="p3">(<span class="default"><a href="quin_1.html#K2" title="QUIN_1:func.2">delta</a> <span class="p4">(<span class="default"><a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p5">(<span class="default"><a href="euclid.html#K11" title="EUCLID:func.11">sqr</a> <span class="p0">(<span class="default"><font color="Maroon" title="c9">T</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c8">S</font></span>)</span></span>)</span></span>)</span>,<span class="p4">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p5"><a href="euclid_2.html#K2" title="EUCLID_2:func.2">|(</a><span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c6">t</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c5">s</font></span>)</span>,<span class="p0">(<span class="default"><font color="Maroon" title="c5">s</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c7">o</font></span>)</span></span><a href="euclid_2.html#K2" title="EUCLID_2:func.2">)|</a></span></span>)</span>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p0">(<span class="default"><a href="euclid.html#K11" title="EUCLID:func.11">sqr</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">S</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c10">O</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p5">(<span class="default"><font color="Maroon" title="c3">r</font> <a href="square_1.html#K3" title="SQUARE_1:func.3">^2</a> </span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</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="euclid.html#K11" title="EUCLID:func.11">sqr</a> <span class="p4">(<span class="default"><font color="Maroon" title="c9">T</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c8">S</font></span>)</span></span>)</span></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"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p3">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p4"><a href="euclid_2.html#K2" title="EUCLID_2:func.2">|(</a><span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c6">t</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c5">s</font></span>)</span>,<span class="p5">(<span class="default"><font color="Maroon" title="c5">s</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c7">o</font></span>)</span></span><a href="euclid_2.html#K2" title="EUCLID_2:func.2">)|</a></span></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p2">(<span class="default"><a href="square_1.html#K7" title="SQUARE_1:func.7">sqrt</a> <span class="p3">(<span class="default"><a href="quin_1.html#K2" title="QUIN_1:func.2">delta</a> <span class="p4">(<span class="default"><a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p5">(<span class="default"><a href="euclid.html#K11" title="EUCLID:func.11">sqr</a> <span class="p0">(<span class="default"><font color="Maroon" title="c9">T</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c8">S</font></span>)</span></span>)</span></span>)</span>,<span class="p4">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p5"><a href="euclid_2.html#K2" title="EUCLID_2:func.2">|(</a><span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c6">t</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c5">s</font></span>)</span>,<span class="p0">(<span class="default"><font color="Maroon" title="c5">s</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c7">o</font></span>)</span></span><a href="euclid_2.html#K2" title="EUCLID_2:func.2">)|</a></span></span>)</span>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p0">(<span class="default"><a href="euclid.html#K11" title="EUCLID:func.11">sqr</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">S</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c10">O</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p5">(<span class="default"><font color="Maroon" title="c3">r</font> <a href="square_1.html#K3" title="SQUARE_1:func.3">^2</a> </span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</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="euclid.html#K11" title="EUCLID:func.11">sqr</a> <span class="p4">(<span class="default"><font color="Maroon" title="c9">T</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c8">S</font></span>)</span></span>)</span></span>)</span></span>)</span> or <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="euclid.html#K11" title="EUCLID:func.11">sqr</a> <span class="p3">(<span class="default"><font color="Maroon" title="c8">S</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c10">O</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">r</font> <a href="square_1.html#K3" title="SQUARE_1:func.3">^2</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 </span><b>by </b><i><a class="txt" href="brouwer.html#E17:13"><i><font color="Green" title="E20">A14</font></i></a>, <a class="txt" href="brouwer.html#E22:13"><i><font color="Green" title="E23">A17</font></i></a>, <a class="txt" href="brouwer.html#E23:13"><i><font color="Green" title="E24">A18</font></i></a>, <a class="ref" href="real_2.html#T106" title="REAL_2:th.106">REAL_2:106</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="13_10_1"><b>suppose </b></a><a NAME="E1:13_10_1"/>
<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="euclid.html#K11" title="EUCLID:func.11">sqr</a> <span class="p3">(<span class="default"><font color="Maroon" title="c8">S</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c10">O</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">r</font> <a href="square_1.html#K3" title="SQUARE_1:func.3">^2</a> </span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ;<br/><div class="add"><a NAME="E2:13_10_1"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p3">(<span class="default"><a href="euclid.html#K11" title="EUCLID:func.11">sqr</a> <span class="p4">(<span class="default"><font color="Maroon" title="c8">S</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c10">O</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">r</font> <a href="square_1.html#K3" title="SQUARE_1:func.3">^2</a> </span>)</span></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</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="euclid.html#K11" title="EUCLID:func.11">sqr</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">T</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c8">S</font></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>by </b><i><a class="txt" href="brouwer.html#E13:13"><i><font color="Green" title="E17">A11</font></i></a>, <a class="ref" href="real_2.html#T128" title="REAL_2:th.128">REAL_2:128</a></i>;<br/><b>hence </b><a NAME="E3:13_10_1"/>
 <a href="brouwer.html#K3" title="BROUWER:func.3">HC</a> <font color="Maroon" title="c5">s</font>,<font color="Maroon" title="c6">t</font>,<font color="Maroon" title="c7">o</font>,<font color="Maroon" title="c3">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">a</font></span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <font color="Maroon" title="c5">s</font></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <font color="Maroon" title="c6">t</font></span>)</span>
 <b>by </b><i><a class="txt" href="brouwer.html#E9:13"><i><font color="Green" title="E13">A7</font></i></a>, <a class="txt" href="brouwer.html#E10:13"><i><font color="Green" title="E14">A8</font></i></a>, <a class="txt" href="brouwer.html#E39:13"><i><font color="Green" title="E30">A24</font></i></a>, <a class="txt" href="brouwer.html#E40:13"><i><font color="Green" title="E31">A25</font></i></a>, <a class="txt" href="brouwer.html#E42:13"><i><font color="Green" title="E32">A26</font></i></a>, <a class="txt" href="brouwer.html#E44:13"><i><font color="Green" title="E34">A28</font></i></a>, <a class="ref" href="xreal_1.html#T135" title="XREAL_1:th.135">XREAL_1:135</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="13_10_2"><b>suppose </b></a><a NAME="E1:13_10_2"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p3">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p4"><a href="euclid_2.html#K2" title="EUCLID_2:func.2">|(</a><span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c6">t</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c5">s</font></span>)</span>,<span class="p5">(<span class="default"><font color="Maroon" title="c5">s</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c7">o</font></span>)</span></span><a href="euclid_2.html#K2" title="EUCLID_2:func.2">)|</a></span></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default"><a href="square_1.html#K7" title="SQUARE_1:func.7">sqrt</a> <span class="p3">(<span class="default"><a href="quin_1.html#K2" title="QUIN_1:func.2">delta</a> <span class="p4">(<span class="default"><a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p5">(<span class="default"><a href="euclid.html#K11" title="EUCLID:func.11">sqr</a> <span class="p0">(<span class="default"><font color="Maroon" title="c9">T</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c8">S</font></span>)</span></span>)</span></span>)</span>,<span class="p4">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p5"><a href="euclid_2.html#K2" title="EUCLID_2:func.2">|(</a><span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c6">t</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c5">s</font></span>)</span>,<span class="p0">(<span class="default"><font color="Maroon" title="c5">s</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c7">o</font></span>)</span></span><a href="euclid_2.html#K2" title="EUCLID_2:func.2">)|</a></span></span>)</span>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p0">(<span class="default"><a href="euclid.html#K11" title="EUCLID:func.11">sqr</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">S</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c10">O</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p5">(<span class="default"><font color="Maroon" title="c3">r</font> <a href="square_1.html#K3" title="SQUARE_1:func.3">^2</a> </span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</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="euclid.html#K11" title="EUCLID:func.11">sqr</a> <span class="p4">(<span class="default"><font color="Maroon" title="c9">T</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c8">S</font></span>)</span></span>)</span></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"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p3">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p4"><a href="euclid_2.html#K2" title="EUCLID_2:func.2">|(</a><span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c6">t</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c5">s</font></span>)</span>,<span class="p5">(<span class="default"><font color="Maroon" title="c5">s</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c7">o</font></span>)</span></span><a href="euclid_2.html#K2" title="EUCLID_2:func.2">)|</a></span></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p2">(<span class="default"><a href="square_1.html#K7" title="SQUARE_1:func.7">sqrt</a> <span class="p3">(<span class="default"><a href="quin_1.html#K2" title="QUIN_1:func.2">delta</a> <span class="p4">(<span class="default"><a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p5">(<span class="default"><a href="euclid.html#K11" title="EUCLID:func.11">sqr</a> <span class="p0">(<span class="default"><font color="Maroon" title="c9">T</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c8">S</font></span>)</span></span>)</span></span>)</span>,<span class="p4">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p5"><a href="euclid_2.html#K2" title="EUCLID_2:func.2">|(</a><span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c6">t</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c5">s</font></span>)</span>,<span class="p0">(<span class="default"><font color="Maroon" title="c5">s</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c7">o</font></span>)</span></span><a href="euclid_2.html#K2" title="EUCLID_2:func.2">)|</a></span></span>)</span>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p0">(<span class="default"><a href="euclid.html#K11" title="EUCLID:func.11">sqr</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">S</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c10">O</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p5">(<span class="default"><font color="Maroon" title="c3">r</font> <a href="square_1.html#K3" title="SQUARE_1:func.3">^2</a> </span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</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="euclid.html#K11" title="EUCLID:func.11">sqr</a> <span class="p4">(<span class="default"><font color="Maroon" title="c9">T</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c8">S</font></span>)</span></span>)</span></span>)</span></span>)</span>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:13_10_2"/>
 <a href="brouwer.html#K3" title="BROUWER:func.3">HC</a> <font color="Maroon" title="c5">s</font>,<font color="Maroon" title="c6">t</font>,<font color="Maroon" title="c7">o</font>,<font color="Maroon" title="c3">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">a</font></span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <font color="Maroon" title="c5">s</font></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <font color="Maroon" title="c6">t</font></span>)</span>
 <b>by </b><i><a class="txt" href="brouwer.html#E9:13"><i><font color="Green" title="E13">A7</font></i></a>, <a class="txt" href="brouwer.html#E39:13"><i><font color="Green" title="E30">A24</font></i></a>, <a class="txt" href="brouwer.html#E40:13"><i><font color="Green" title="E31">A25</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="13_10_3"><b>suppose </b></a><a NAME="E1:13_10_3"/><i><font color="Green" title="E35">A30</font></i>: 
<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="euclid.html#K11" title="EUCLID:func.11">sqr</a> <span class="p3">(<span class="default"><font color="Maroon" title="c8">S</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c10">O</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">r</font> <a href="square_1.html#K3" title="SQUARE_1:func.3">^2</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ;<br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="13_10_3_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:13_10_3_1"/><i><font color="Green" title="E36">A31</font></i>: 
<font color="Maroon" title="c18">l</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p3">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p4"><a href="euclid_2.html#K2" title="EUCLID_2:func.2">|(</a><span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c6">t</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c5">s</font></span>)</span>,<span class="p5">(<span class="default"><font color="Maroon" title="c5">s</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c7">o</font></span>)</span></span><a href="euclid_2.html#K2" title="EUCLID_2:func.2">)|</a></span></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default"><a href="square_1.html#K7" title="SQUARE_1:func.7">sqrt</a> <span class="p3">(<span class="default"><a href="quin_1.html#K2" title="QUIN_1:func.2">delta</a> <span class="p4">(<span class="default"><a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p5">(<span class="default"><a href="euclid.html#K11" title="EUCLID:func.11">sqr</a> <span class="p0">(<span class="default"><font color="Maroon" title="c9">T</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c8">S</font></span>)</span></span>)</span></span>)</span>,<span class="p4">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p5"><a href="euclid_2.html#K2" title="EUCLID_2:func.2">|(</a><span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c6">t</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c5">s</font></span>)</span>,<span class="p0">(<span class="default"><font color="Maroon" title="c5">s</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c7">o</font></span>)</span></span><a href="euclid_2.html#K2" title="EUCLID_2:func.2">)|</a></span></span>)</span>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p0">(<span class="default"><a href="euclid.html#K11" title="EUCLID:func.11">sqr</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">S</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c10">O</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p5">(<span class="default"><font color="Maroon" title="c3">r</font> <a href="square_1.html#K3" title="SQUARE_1:func.3">^2</a> </span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</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="euclid.html#K11" title="EUCLID:func.11">sqr</a> <span class="p4">(<span class="default"><font color="Maroon" title="c9">T</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c8">S</font></span>)</span></span>)</span></span>)</span></span>)</span>
 ;<br/><div><i><font color="Green" title="E37">A32</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="13_10_3_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:13_10_3_1_1"/>
<font color="Maroon" title="c18">l</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ;<br/><b>then </b> <a href="brouwer.html#K3" title="BROUWER:func.3">HC</a> <font color="Maroon" title="c5">s</font>,<font color="Maroon" title="c6">t</font>,<font color="Maroon" title="c7">o</font>,<font color="Maroon" title="c3">r</font> = 
<font color="Maroon" title="c5">s</font> <a href="euclid.html#K17" title="EUCLID:func.17">+</a> <span class="p1">(<span class="default"><a href="euclid.html#K16" title="EUCLID:func.16">0.REAL</a> <font color="Maroon" title="c4">n</font></span>)</span>
<b>by </b><i><a class="txt" href="brouwer.html#E11:13"><i><font color="Green" title="E15">A9</font></i></a>, <a class="ref" href="euclid.html#T33" title="EUCLID:th.33">EUCLID:33</a></i>
<br/>.= 
<font color="Maroon" title="c5">s</font>
<b>by </b><i><a class="ref" href="euclid.html#T31" title="EUCLID:th.31">EUCLID:31</a></i>
;<br/><b>hence </b><a NAME="E3:13_10_3_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="brouwer.html#E2:13"><i><font color="Green" title="E8">A2</font></i></a>, <a class="txt" href="brouwer.html#E3:13"><i><font color="Green" title="E9">A3</font></i></a>, <a class="txt" href="brouwer.html#E4:13"><i><font color="Green" title="E10">A4</font></i></a>, <a class="ref" href="brouwer.html#D3" target="_self" title="BROUWER:def.3">Def3</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="13_10_3_1_2"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:13_10_3_1_2"/>
(  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1"><a href="euclid_2.html#K2" title="EUCLID_2:func.2">|(</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">t</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c5">s</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c5">s</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c7">o</font></span>)</span></span><a href="euclid_2.html#K2" title="EUCLID_2:func.2">)|</a></span> or 2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1"><a href="euclid_2.html#K2" title="EUCLID_2:func.2">|(</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">t</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c5">s</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c5">s</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c7">o</font></span>)</span></span><a href="euclid_2.html#K2" title="EUCLID_2:func.2">)|</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="13_10_3_1_2_1"><b>suppose </b></a><a NAME="E1:13_10_3_1_2_1"/><i><font color="Green" title="E38">A33</font></i>: 
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1"><a href="euclid_2.html#K2" title="EUCLID_2:func.2">|(</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">t</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c5">s</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c5">s</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c7">o</font></span>)</span></span><a href="euclid_2.html#K2" title="EUCLID_2:func.2">)|</a></span>
 ;<br/><div class="add"><a NAME="E2:13_10_3_1_2_1"/><b>then </b><i><font color="Green" title="E39">A34</font></i>: 
<span class="p1">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 2</span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2"><a href="euclid_2.html#K2" title="EUCLID_2:func.2">|(</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c6">t</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c5">s</font></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon" title="c5">s</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c7">o</font></span>)</span></span><a href="euclid_2.html#K2" title="EUCLID_2:func.2">)|</a></span></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>by </b><i><a class="ref" href="xreal_1.html#T134" title="XREAL_1:th.134">XREAL_1:134</a></i>;<br/><a NAME="E3:13_10_3_1_2_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p3">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p4"><a href="euclid_2.html#K2" title="EUCLID_2:func.2">|(</a><span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c6">t</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c5">s</font></span>)</span>,<span class="p5">(<span class="default"><font color="Maroon" title="c5">s</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c7">o</font></span>)</span></span><a href="euclid_2.html#K2" title="EUCLID_2:func.2">)|</a></span></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default"><a href="square_1.html#K7" title="SQUARE_1:func.7">sqrt</a> <span class="p3">(<span class="default"><a href="quin_1.html#K2" title="QUIN_1:func.2">delta</a> <span class="p4">(<span class="default"><a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p5">(<span class="default"><a href="euclid.html#K11" title="EUCLID:func.11">sqr</a> <span class="p0">(<span class="default"><font color="Maroon" title="c9">T</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c8">S</font></span>)</span></span>)</span></span>)</span>,<span class="p4">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p5"><a href="euclid_2.html#K2" title="EUCLID_2:func.2">|(</a><span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c6">t</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c5">s</font></span>)</span>,<span class="p0">(<span class="default"><font color="Maroon" title="c5">s</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c7">o</font></span>)</span></span><a href="euclid_2.html#K2" title="EUCLID_2:func.2">)|</a></span></span>)</span>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p0">(<span class="default"><a href="euclid.html#K11" title="EUCLID:func.11">sqr</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">S</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c10">O</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p5">(<span class="default"><font color="Maroon" title="c3">r</font> <a href="square_1.html#K3" title="SQUARE_1:func.3">^2</a> </span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</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="euclid.html#K11" title="EUCLID:func.11">sqr</a> <span class="p4">(<span class="default"><font color="Maroon" title="c9">T</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c8">S</font></span>)</span></span>)</span></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"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p3">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p4"><a href="euclid_2.html#K2" title="EUCLID_2:func.2">|(</a><span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c6">t</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c5">s</font></span>)</span>,<span class="p5">(<span class="default"><font color="Maroon" title="c5">s</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c7">o</font></span>)</span></span><a href="euclid_2.html#K2" title="EUCLID_2:func.2">)|</a></span></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p3"><a href="euclid_2.html#K2" title="EUCLID_2:func.2">|(</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c6">t</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c5">s</font></span>)</span>,<span class="p4">(<span class="default"><font color="Maroon" title="c5">s</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c7">o</font></span>)</span></span><a href="euclid_2.html#K2" title="EUCLID_2:func.2">)|</a></span></span>)</span></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</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="euclid.html#K11" title="EUCLID:func.11">sqr</a> <span class="p4">(<span class="default"><font color="Maroon" title="c9">T</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c8">S</font></span>)</span></span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="brouwer.html#E25:13"><i><font color="Green" title="E26">A20</font></i></a>, <a class="txt" href="brouwer.html#E1:13_10_3"><i><font color="Green" title="E35">A30</font></i></a>, <a class="txt" href="brouwer.html#E1:13_10_3_1_2_1"><i><font color="Green" title="E38">A33</font></i></a>, <a class="ref" href="square_1.html#T89" title="SQUARE_1:th.89">SQUARE_1:89</a></i>;<br/><b>hence </b><a NAME="E4:13_10_3_1_2_1"/>
contradiction
 <b>by </b><i><a class="txt" href="brouwer.html#E10:13"><i><font color="Green" title="E14">A8</font></i></a>, <a class="txt" href="brouwer.html#E43:13"><i><font color="Green" title="E33">A27</font></i></a>, <a class="txt" href="brouwer.html#E1:13_10_3_1"><i><font color="Green" title="E36">A31</font></i></a>, <a class="txt" href="brouwer.html#E2:13_10_3_1_2_1"><i><font color="Green" title="E39">A34</font></i></a>, <a class="ref" href="real_2.html#T128" title="REAL_2:th.128">REAL_2:128</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="13_10_3_1_2_2"><b>suppose </b></a><a NAME="E1:13_10_3_1_2_2"/>
2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1"><a href="euclid_2.html#K2" title="EUCLID_2:func.2">|(</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">t</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c5">s</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c5">s</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c7">o</font></span>)</span></span><a href="euclid_2.html#K2" title="EUCLID_2:func.2">)|</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ;<br/><div class="add"><b>then </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p3">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p4"><a href="euclid_2.html#K2" title="EUCLID_2:func.2">|(</a><span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c6">t</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c5">s</font></span>)</span>,<span class="p5">(<span class="default"><font color="Maroon" title="c5">s</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c7">o</font></span>)</span></span><a href="euclid_2.html#K2" title="EUCLID_2:func.2">)|</a></span></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default"><a href="square_1.html#K7" title="SQUARE_1:func.7">sqrt</a> <span class="p3">(<span class="default"><a href="quin_1.html#K2" title="QUIN_1:func.2">delta</a> <span class="p4">(<span class="default"><a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p5">(<span class="default"><a href="euclid.html#K11" title="EUCLID:func.11">sqr</a> <span class="p0">(<span class="default"><font color="Maroon" title="c9">T</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c8">S</font></span>)</span></span>)</span></span>)</span>,<span class="p4">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p5"><a href="euclid_2.html#K2" title="EUCLID_2:func.2">|(</a><span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c6">t</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c5">s</font></span>)</span>,<span class="p0">(<span class="default"><font color="Maroon" title="c5">s</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c7">o</font></span>)</span></span><a href="euclid_2.html#K2" title="EUCLID_2:func.2">)|</a></span></span>)</span>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p0">(<span class="default"><a href="euclid.html#K11" title="EUCLID:func.11">sqr</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">S</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c10">O</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p5">(<span class="default"><font color="Maroon" title="c3">r</font> <a href="square_1.html#K3" title="SQUARE_1:func.3">^2</a> </span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</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="euclid.html#K11" title="EUCLID:func.11">sqr</a> <span class="p4">(<span class="default"><font color="Maroon" title="c9">T</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c8">S</font></span>)</span></span>)</span></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p3">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p4"><a href="euclid_2.html#K2" title="EUCLID_2:func.2">|(</a><span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c6">t</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c5">s</font></span>)</span>,<span class="p5">(<span class="default"><font color="Maroon" title="c5">s</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c7">o</font></span>)</span></span><a href="euclid_2.html#K2" title="EUCLID_2:func.2">)|</a></span></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p3">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p4"><a href="euclid_2.html#K2" title="EUCLID_2:func.2">|(</a><span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c6">t</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c5">s</font></span>)</span>,<span class="p5">(<span class="default"><font color="Maroon" title="c5">s</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c7">o</font></span>)</span></span><a href="euclid_2.html#K2" title="EUCLID_2:func.2">)|</a></span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</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="euclid.html#K11" title="EUCLID:func.11">sqr</a> <span class="p4">(<span class="default"><font color="Maroon" title="c9">T</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c8">S</font></span>)</span></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="brouwer.html#E25:13"><i><font color="Green" title="E26">A20</font></i></a>, <a class="txt" href="brouwer.html#E1:13_10_3"><i><font color="Green" title="E35">A30</font></i></a>, <a class="ref" href="square_1.html#T90" title="SQUARE_1:th.90">SQUARE_1:90</a></i>
<br/>.= 
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 

;<br/><b>hence </b><a NAME="E3:13_10_3_1_2_2"/>
contradiction
 <b>by </b><i><a class="txt" href="brouwer.html#E1:13_10_3_1"><i><font color="Green" title="E36">A31</font></i></a>, <a class="txt" href="brouwer.html#E2:13_10_3_1"><i><font color="Green" title="E37">A32</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E3:13_10_3"/>
 <a href="brouwer.html#K3" title="BROUWER:func.3">HC</a> <font color="Maroon" title="c5">s</font>,<font color="Maroon" title="c6">t</font>,<font color="Maroon" title="c7">o</font>,<font color="Maroon" title="c3">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">a</font></span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <font color="Maroon" title="c5">s</font></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <font color="Maroon" title="c6">t</font></span>)</span>
 <b>by </b><i><a class="txt" href="brouwer.html#E9:13"><i><font color="Green" title="E13">A7</font></i></a>, <a class="txt" href="brouwer.html#E39:13"><i><font color="Green" title="E30">A24</font></i></a>, <a class="txt" href="brouwer.html#E40:13"><i><font color="Green" title="E31">A25</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

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