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<b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>let </b><font color="Maroon">r</font> be  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ;<br/><b>let </b><font color="Maroon">y</font> be   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> <font color="Maroon">n</font></span>)</span>;<br/><b>let </b><font color="Maroon">x</font> be   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> <font color="Maroon">n</font></span>)</span>;<br/><b>let </b><font color="Maroon">z</font> be   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> <font color="Maroon">n</font></span>)</span>;<br/>









<b>assume </b><b>that </b><br/><a NAME="E1:50"/><i><font color="Green">E42</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="topreal9.html#K3" target="_self">Sphere</a> <font color="Maroon">x</font>,<font color="Maroon">r</font>
 <b>and </b><br/><a NAME="E2:50"/><i><font color="Green">E44</font></i>: 
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a>  <a href="topreal9.html#K3" target="_self">Sphere</a> <font color="Maroon">x</font>,<font color="Maroon">r</font>
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="50_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:50_1"/>
( <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">z</font> or <font color="Maroon">y</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">z</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="50_1_1"><b>suppose </b></a><a NAME="E1:50_1_1"/>
<font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">z</font>
 ;<br/><div class="add"><a NAME="E2:50_1_1"/><b>then </b>
(  <a href="topreal1.html#K3">LSeg</a> <font color="Maroon">y</font>,<font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">y</font></span><a href="tarski.html#K1">}</a></span> &amp; <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">z</font></span><a href="tarski.html#K2">}</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">y</font></span><a href="tarski.html#K1">}</a></span> )
 <b>by </b><i><a class="ref" href="enumset1.html#T69">ENUMSET1:69</a>, <a class="ref" href="topreal1.html#T7">TOPREAL1:7</a></i>;<br/><a NAME="E3:50_1_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <font color="Maroon">y</font>,<font color="Maroon">z</font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">z</font></span><a href="tarski.html#K2">}</a></span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i><a class="ref" href="xboole_1.html#T37">XBOOLE_1:37</a></i>;<br/><b>hence </b><a NAME="E4:50_1_1"/>
<span class="p1">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <font color="Maroon">y</font>,<font color="Maroon">z</font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">z</font></span><a href="tarski.html#K2">}</a></span> <a href="tarski.html#R1">c=</a>  <a href="topreal9.html#K1" target="_self">Ball</a> <font color="Maroon">x</font>,<font color="Maroon">r</font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T2">XBOOLE_1:2</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="50_1_2"><b>suppose </b></a><a NAME="E1:50_1_2"/><i><font color="Green">E46</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">z</font>
 ;<br/><div class="add"><b>let </b><font color="Maroon">a</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/><b>assume </b><a NAME="E2:50_1_2"/><i><font color="Green">E67</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <font color="Maroon">y</font>,<font color="Maroon">z</font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">z</font></span><a href="tarski.html#K2">}</a></span>
 ;<br/><b>then reconsider </b><font color="Maroon">R</font> = <font color="Maroon">a</font> as   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> <font color="Maroon">n</font></span>)</span> ;<br/><a NAME="E4:50_1_2"/><i><font color="Green">E68</font></i>: 
<font color="Maroon">R</font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3">LSeg</a> <font color="Maroon">y</font>,<font color="Maroon">z</font>
 <b>by </b><i>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/><b>then consider </b><font color="Maroon">l</font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E6:50_1_2"/><i><font color="Green">E69</font></i>: 
0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">l</font>
 <b>and </b><br/><a NAME="E7:50_1_2"/><i><font color="Green">E70</font></i>: 
<font color="Maroon">l</font> <a href="xxreal_0.html#R1">&lt;=</a> 1
 <b>and </b><br/><a NAME="E8:50_1_2"/><i><font color="Green">E71</font></i>: 
<font color="Maroon">R</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">l</font></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">y</font></span>)</span> <a href="euclid.html#K17">+</a> <span class="p1">(<span class="default"><font color="Maroon">l</font> <a href="euclid.html#K18">*</a> <font color="Maroon">z</font></span>)</span>
 <b>by </b><i><a class="ref" href="jgraph_1.html#T52">JGRAPH_1:52</a></i>;<br/><b>set </b><font color="Maroon">l1</font> = 1 <a href="real_1.html#K5">-</a> <font color="Maroon">l</font>;<br/><b>set </b><font color="Maroon">X</font> = <span class="p1"><a href="euclid_2.html#K2">|(</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">y</font> <a href="euclid.html#K20">-</a> <font color="Maroon">x</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">z</font> <a href="euclid.html#K20">-</a> <font color="Maroon">x</font></span>)</span></span><a href="euclid_2.html#K2">)|</a></span>;<br/><a NAME="E9:50_1_2"/><i><font color="Green">E72</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> <font color="Maroon">n</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="euclid.html#K1">REAL</a> <font color="Maroon">n</font>
 <b>by </b><i><a class="ref" href="euclid.html#T25">EUCLID:25</a></i>;<br/><b>then reconsider </b><font color="Maroon">xf</font> = <font color="Maroon">x</font>, <font color="Maroon">yf</font> = <font color="Maroon">y</font>, <font color="Maroon">zf</font> = <font color="Maroon">z</font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">n</font> <a href="finseq_2.html#K4">-tuples_on</a> <a href="numbers.html#K1">REAL</a>  ;<br/><b>reconsider </b><font color="Maroon">y1</font> = <font color="Maroon">y</font> <a href="euclid.html#K20">-</a> <font color="Maroon">x</font>, <font color="Maroon">z1</font> = <font color="Maroon">z</font> <a href="euclid.html#K20">-</a> <font color="Maroon">x</font> as    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K1">REAL</a>  <b>by </b><i><a class="ref" href="euclid.html#T27">EUCLID:27</a></i>;<br/><b>consider </b><font color="Maroon">Rf</font> being    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K1">REAL</a> <b> such that </b><br/><a NAME="E13:50_1_2"/><i><font color="Green">E73</font></i>: 
<font color="Maroon">R</font> <a href="euclid.html#K20">-</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">Rf</font>
 <b>and </b><br/><a NAME="E14:50_1_2"/><i><font color="Green">E74</font></i>: 
<span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">R</font> <a href="euclid.html#K20">-</a> <font color="Maroon">x</font></span>)</span></span><a href="toprns_1.html#K5">.|</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="euclid.html#K12">|.</a><span class="default"><font color="Maroon">Rf</font></span><a href="euclid.html#K12">.|</a></span>
 <b>by </b><i><a class="ref" href="toprns_1.html#D6">TOPRNS_1:def 6</a></i>;<br/><a NAME="E15:50_1_2"/><i><font color="Green">E75</font></i>: 
 <a href="rvsum_1.html#K15">Sum</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K11">sqr</a> <font color="Maroon">Rf</font></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 <b>by </b><i><a class="ref" href="rvsum_1.html#T116">RVSUM_1:116</a></i>;<br/><a NAME="E16:50_1_2"/><i><font color="Green">E83</font></i>: 
<span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">y</font> <a href="euclid.html#K20">-</a> <font color="Maroon">x</font></span>)</span></span><a href="toprns_1.html#K5">.|</a></span> <a href="square_1.html#K5">^2</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">r</font> <a href="square_1.html#K3">^2</a> 
 <b>by </b><i>, </i>;<br/><a NAME="E17:50_1_2"/><i><font color="Green">E84</font></i>: 
<span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">z</font> <a href="euclid.html#K20">-</a> <font color="Maroon">x</font></span>)</span></span><a href="toprns_1.html#K5">.|</a></span> <a href="square_1.html#K5">^2</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">r</font> <a href="square_1.html#K3">^2</a> 
 <b>by </b><i>, </i>;<br/><b>reconsider </b><font color="Maroon">W1</font> = <span class="p1">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">l</font></span>)</span> <a href="euclid.html#K18">*</a> <span class="p1">(<span class="default"><font color="Maroon">y</font> <a href="euclid.html#K20">-</a> <font color="Maroon">x</font></span>)</span>, <font color="Maroon">W2</font> = <font color="Maroon">l</font> <a href="euclid.html#K18">*</a> <span class="p1">(<span class="default"><font color="Maroon">z</font> <a href="euclid.html#K20">-</a> <font color="Maroon">x</font></span>)</span> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">n</font> <a href="finseq_2.html#K4">-tuples_on</a> <a href="numbers.html#K1">REAL</a>  <b>by </b><i/>;<br/><a NAME="E19:50_1_2"/><i><font color="Green">E85</font></i>: 
<font color="Maroon">y</font> <a href="euclid.html#K20">-</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">yf</font> <a href="rvsum_1.html#K8">-</a> <font color="Maroon">xf</font>
 <b>by </b><i><a class="ref" href="euclid.html#D13">EUCLID:def 13</a></i>;<br/><a NAME="E20:50_1_2"/><b>then </b><i><font color="Green">E86</font></i>: 
<span class="p1">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">l</font></span>)</span> <a href="euclid.html#K18">*</a> <span class="p1">(<span class="default"><font color="Maroon">y</font> <a href="euclid.html#K20">-</a> <font color="Maroon">x</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">l</font></span>)</span> <a href="rvsum_1.html#K10">*</a> <span class="p1">(<span class="default"><font color="Maroon">yf</font> <a href="rvsum_1.html#K8">-</a> <font color="Maroon">xf</font></span>)</span>
 <b>by </b><i><a class="ref" href="euclid.html#D11">EUCLID:def 11</a></i>;<br/><a NAME="E21:50_1_2"/><i><font color="Green">E119</font></i>: 
<font color="Maroon">z</font> <a href="euclid.html#K20">-</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">zf</font> <a href="rvsum_1.html#K8">-</a> <font color="Maroon">xf</font>
 <b>by </b><i><a class="ref" href="euclid.html#D13">EUCLID:def 13</a></i>;<br/><a NAME="E22:50_1_2"/><b>then </b><i><font color="Green">E120</font></i>: 
<font color="Maroon">l</font> <a href="euclid.html#K18">*</a> <span class="p1">(<span class="default"><font color="Maroon">z</font> <a href="euclid.html#K20">-</a> <font color="Maroon">x</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">l</font> <a href="rvsum_1.html#K10">*</a> <span class="p1">(<span class="default"><font color="Maroon">zf</font> <a href="rvsum_1.html#K8">-</a> <font color="Maroon">xf</font></span>)</span>
 <b>by </b><i><a class="ref" href="euclid.html#D11">EUCLID:def 11</a></i>;<br/><i><font color="Green">E121</font></i>: <font color="Maroon">W1</font> <a href="rvsum_1.html#K4">+</a> <font color="Maroon">W2</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">l</font></span>)</span> <a href="euclid.html#K18">*</a> <span class="p2">(<span class="default"><font color="Maroon">y</font> <a href="euclid.html#K20">-</a> <font color="Maroon">x</font></span>)</span></span>)</span> <a href="euclid.html#K17">+</a> <span class="p1">(<span class="default"><font color="Maroon">l</font> <a href="euclid.html#K18">*</a> <span class="p2">(<span class="default"><font color="Maroon">z</font> <a href="euclid.html#K20">-</a> <font color="Maroon">x</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="euclid.html#D10">EUCLID:def 10</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">l</font></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">y</font></span>)</span> <a href="euclid.html#K20">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">l</font></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">x</font></span>)</span></span>)</span> <a href="euclid.html#K17">+</a> <span class="p1">(<span class="default"><font color="Maroon">l</font> <a href="euclid.html#K18">*</a> <span class="p2">(<span class="default"><font color="Maroon">z</font> <a href="euclid.html#K20">-</a> <font color="Maroon">x</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="euclid.html#T53">EUCLID:53</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">l</font></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">y</font></span>)</span> <a href="euclid.html#K20">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">l</font></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">x</font></span>)</span></span>)</span> <a href="euclid.html#K17">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">l</font> <a href="euclid.html#K18">*</a> <font color="Maroon">z</font></span>)</span> <a href="euclid.html#K20">-</a> <span class="p2">(<span class="default"><font color="Maroon">l</font> <a href="euclid.html#K18">*</a> <font color="Maroon">x</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="euclid.html#T53">EUCLID:53</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">l</font></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">y</font></span>)</span> <a href="euclid.html#K20">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">l</font></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">x</font></span>)</span></span>)</span> <a href="euclid.html#K17">+</a> <span class="p2">(<span class="default"><font color="Maroon">l</font> <a href="euclid.html#K18">*</a> <font color="Maroon">z</font></span>)</span></span>)</span> <a href="euclid.html#K20">-</a> <span class="p1">(<span class="default"><font color="Maroon">l</font> <a href="euclid.html#K18">*</a> <font color="Maroon">x</font></span>)</span>
<b>by </b><i><a class="ref" href="euclid.html#T49">EUCLID:49</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">l</font></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">y</font></span>)</span> <a href="euclid.html#K17">+</a> <span class="p3">(<span class="default"><font color="Maroon">l</font> <a href="euclid.html#K18">*</a> <font color="Maroon">z</font></span>)</span></span>)</span> <a href="euclid.html#K20">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">l</font></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">x</font></span>)</span></span>)</span> <a href="euclid.html#K20">-</a> <span class="p1">(<span class="default"><font color="Maroon">l</font> <a href="euclid.html#K18">*</a> <font color="Maroon">x</font></span>)</span>
<b>by </b><i><a class="ref" href="jordan2c.html#T9">JORDAN2C:9</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">l</font></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">y</font></span>)</span> <a href="euclid.html#K17">+</a> <span class="p2">(<span class="default"><font color="Maroon">l</font> <a href="euclid.html#K18">*</a> <font color="Maroon">z</font></span>)</span></span>)</span> <a href="euclid.html#K20">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">l</font></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">x</font></span>)</span> <a href="euclid.html#K17">+</a> <span class="p2">(<span class="default"><font color="Maroon">l</font> <a href="euclid.html#K18">*</a> <font color="Maroon">x</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="euclid.html#T50">EUCLID:50</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">l</font></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">y</font></span>)</span> <a href="euclid.html#K17">+</a> <span class="p2">(<span class="default"><font color="Maroon">l</font> <a href="euclid.html#K18">*</a> <font color="Maroon">z</font></span>)</span></span>)</span> <a href="euclid.html#K20">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="euclid.html#K18">*</a> <font color="Maroon">x</font></span>)</span> <a href="euclid.html#K20">-</a> <span class="p3">(<span class="default"><font color="Maroon">l</font> <a href="euclid.html#K18">*</a> <font color="Maroon">x</font></span>)</span></span>)</span> <a href="euclid.html#K17">+</a> <span class="p2">(<span class="default"><font color="Maroon">l</font> <a href="euclid.html#K18">*</a> <font color="Maroon">x</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="euclid.html#T54">EUCLID:54</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">l</font></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">y</font></span>)</span> <a href="euclid.html#K17">+</a> <span class="p2">(<span class="default"><font color="Maroon">l</font> <a href="euclid.html#K18">*</a> <font color="Maroon">z</font></span>)</span></span>)</span> <a href="euclid.html#K20">-</a> <span class="p1">(<span class="default">1 <a href="euclid.html#K18">*</a> <font color="Maroon">x</font></span>)</span>
<b>by </b><i><a class="ref" href="euclid.html#T52">EUCLID:52</a></i>
<br/>.= 
<font color="Maroon">Rf</font>
<b>by </b><i>, , <a class="ref" href="euclid.html#T33">EUCLID:33</a></i>
;<br/><span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">R</font> <a href="euclid.html#K20">-</a> <font color="Maroon">x</font></span>)</span></span><a href="toprns_1.html#K5">.|</a></span> <a href="square_1.html#K5">^2</a>  = 
 <a href="rvsum_1.html#K15">Sum</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K11">sqr</a> <font color="Maroon">Rf</font></span>)</span>
<b>by </b><i>, <a class="ref" href="topreal9.html#T1" target="_self">Th1</a>, <a class="ref" href="square_1.html#D4">SQUARE_1:def 4</a></i>
<br/>.= 
 <a href="rvsum_1.html#K15">Sum</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="rvsum_1.html#K12">sqr</a> <font color="Maroon">W1</font></span>)</span> <a href="rvsum_1.html#K4">+</a> <span class="p3">(<span class="default">2 <a href="rvsum_1.html#K10">*</a> <span class="p4">(<span class="default"><a href="rvsum_1.html#K14">mlt</a> <font color="Maroon">W1</font>,<font color="Maroon">W2</font></span>)</span></span>)</span></span>)</span> <a href="rvsum_1.html#K4">+</a> <span class="p2">(<span class="default"><a href="rvsum_1.html#K12">sqr</a> <font color="Maroon">W2</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="topreal9.html#T5" target="_self">Th5</a>, <a class="ref" href="rvsum_1.html#T98">RVSUM_1:98</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="rvsum_1.html#K12">sqr</a> <font color="Maroon">W1</font></span>)</span> <a href="rvsum_1.html#K4">+</a> <span class="p3">(<span class="default">2 <a href="rvsum_1.html#K10">*</a> <span class="p4">(<span class="default"><a href="rvsum_1.html#K14">mlt</a> <font color="Maroon">W1</font>,<font color="Maroon">W2</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <span class="p2">(<span class="default"><a href="rvsum_1.html#K12">sqr</a> <font color="Maroon">W2</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="rvsum_1.html#T119">RVSUM_1:119</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <span class="p3">(<span class="default"><a href="rvsum_1.html#K12">sqr</a> <font color="Maroon">W1</font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <span class="p3">(<span class="default">2 <a href="rvsum_1.html#K10">*</a> <span class="p4">(<span class="default"><a href="rvsum_1.html#K14">mlt</a> <font color="Maroon">W1</font>,<font color="Maroon">W2</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <span class="p2">(<span class="default"><a href="rvsum_1.html#K12">sqr</a> <font color="Maroon">W2</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="rvsum_1.html#T119">RVSUM_1:119</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">l</font></span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span> <a href="rvsum_1.html#K10">*</a> <span class="p4">(<span class="default"><a href="rvsum_1.html#K12">sqr</a> <span class="p5">(<span class="default"><font color="Maroon">yf</font> <a href="rvsum_1.html#K8">-</a> <font color="Maroon">xf</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <span class="p3">(<span class="default">2 <a href="rvsum_1.html#K10">*</a> <span class="p4">(<span class="default"><a href="rvsum_1.html#K14">mlt</a> <font color="Maroon">W1</font>,<font color="Maroon">W2</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <span class="p2">(<span class="default"><a href="rvsum_1.html#K12">sqr</a> <span class="p3">(<span class="default"><font color="Maroon">l</font> <a href="rvsum_1.html#K10">*</a> <span class="p4">(<span class="default"><font color="Maroon">zf</font> <a href="rvsum_1.html#K8">-</a> <font color="Maroon">xf</font></span>)</span></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="topreal9.html#E2:50"><i><font color="Green">E44</font></i></a>, <a class="txt" href="topreal9.html#E1:50_1_2"><i><font color="Green">E46</font></i></a>, <a class="ref" href="rvsum_1.html#T84">RVSUM_1:84</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">l</font></span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span> <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <span class="p4">(<span class="default"><a href="rvsum_1.html#K12">sqr</a> <span class="p5">(<span class="default"><font color="Maroon">yf</font> <a href="rvsum_1.html#K8">-</a> <font color="Maroon">xf</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <span class="p3">(<span class="default">2 <a href="rvsum_1.html#K10">*</a> <span class="p4">(<span class="default"><a href="rvsum_1.html#K14">mlt</a> <font color="Maroon">W1</font>,<font color="Maroon">W2</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <span class="p2">(<span class="default"><a href="rvsum_1.html#K12">sqr</a> <span class="p3">(<span class="default"><font color="Maroon">l</font> <a href="rvsum_1.html#K10">*</a> <span class="p4">(<span class="default"><font color="Maroon">zf</font> <a href="rvsum_1.html#K8">-</a> <font color="Maroon">xf</font></span>)</span></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="rvsum_1.html#T117">RVSUM_1:117</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">l</font></span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span> <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <span class="p4">(<span class="default"><a href="rvsum_1.html#K12">sqr</a> <span class="p5">(<span class="default"><font color="Maroon">yf</font> <a href="rvsum_1.html#K8">-</a> <font color="Maroon">xf</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <span class="p3">(<span class="default">2 <a href="rvsum_1.html#K10">*</a> <span class="p4">(<span class="default"><a href="rvsum_1.html#K14">mlt</a> <font color="Maroon">W1</font>,<font color="Maroon">W2</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">l</font> <a href="square_1.html#K5">^2</a> </span>)</span> <a href="rvsum_1.html#K10">*</a> <span class="p3">(<span class="default"><a href="rvsum_1.html#K12">sqr</a> <span class="p4">(<span class="default"><font color="Maroon">zf</font> <a href="rvsum_1.html#K8">-</a> <font color="Maroon">xf</font></span>)</span></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="rvsum_1.html#T84">RVSUM_1:84</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">l</font></span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span> <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <span class="p4">(<span class="default"><a href="rvsum_1.html#K12">sqr</a> <span class="p5">(<span class="default"><font color="Maroon">yf</font> <a href="rvsum_1.html#K8">-</a> <font color="Maroon">xf</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <span class="p3">(<span class="default">2 <a href="rvsum_1.html#K10">*</a> <span class="p4">(<span class="default"><a href="rvsum_1.html#K14">mlt</a> <font color="Maroon">W1</font>,<font color="Maroon">W2</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">l</font> <a href="square_1.html#K5">^2</a> </span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <span class="p3">(<span class="default"><a href="rvsum_1.html#K12">sqr</a> <span class="p4">(<span class="default"><font color="Maroon">zf</font> <a href="rvsum_1.html#K8">-</a> <font color="Maroon">xf</font></span>)</span></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="rvsum_1.html#T117">RVSUM_1:117</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">l</font></span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span> <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default"><span class="p4"><a href="euclid.html#K12">|.</a><span class="default"><font color="Maroon">y1</font></span><a href="euclid.html#K12">.|</a></span> <a href="square_1.html#K5">^2</a> </span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <span class="p3">(<span class="default">2 <a href="rvsum_1.html#K10">*</a> <span class="p4">(<span class="default"><a href="rvsum_1.html#K14">mlt</a> <font color="Maroon">W1</font>,<font color="Maroon">W2</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">l</font> <a href="square_1.html#K5">^2</a> </span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <span class="p3">(<span class="default"><a href="rvsum_1.html#K12">sqr</a> <span class="p4">(<span class="default"><font color="Maroon">zf</font> <a href="rvsum_1.html#K8">-</a> <font color="Maroon">xf</font></span>)</span></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="topreal9.html#T3" target="_self">Th3</a>, </i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">l</font></span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span> <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default"><span class="p4"><a href="euclid.html#K12">|.</a><span class="default"><font color="Maroon">y1</font></span><a href="euclid.html#K12">.|</a></span> <a href="square_1.html#K5">^2</a> </span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <span class="p3">(<span class="default">2 <a href="rvsum_1.html#K10">*</a> <span class="p4">(<span class="default"><a href="rvsum_1.html#K14">mlt</a> <font color="Maroon">W1</font>,<font color="Maroon">W2</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">l</font> <a href="square_1.html#K5">^2</a> </span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><span class="p3"><a href="euclid.html#K12">|.</a><span class="default"><font color="Maroon">z1</font></span><a href="euclid.html#K12">.|</a></span> <a href="square_1.html#K5">^2</a> </span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="topreal9.html#T4" target="_self">Th4</a>, </i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">l</font></span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p3">(<span class="default"><font color="Maroon">r</font> <a href="square_1.html#K3">^2</a> </span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <span class="p3">(<span class="default">2 <a href="rvsum_1.html#K10">*</a> <span class="p4">(<span class="default"><a href="rvsum_1.html#K14">mlt</a> <font color="Maroon">W1</font>,<font color="Maroon">W2</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">l</font> <a href="square_1.html#K5">^2</a> </span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><span class="p3"><a href="euclid.html#K12">|.</a><span class="default"><font color="Maroon">z1</font></span><a href="euclid.html#K12">.|</a></span> <a href="square_1.html#K5">^2</a> </span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="topreal9.html#T2" target="_self">Th2</a>, <a class="ref" href="toprns_1.html#D6">TOPRNS_1:def 6</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">l</font></span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p3">(<span class="default"><font color="Maroon">r</font> <a href="square_1.html#K3">^2</a> </span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <span class="p3">(<span class="default">2 <a href="rvsum_1.html#K10">*</a> <span class="p4">(<span class="default"><a href="rvsum_1.html#K14">mlt</a> <font color="Maroon">W1</font>,<font color="Maroon">W2</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">l</font> <a href="square_1.html#K5">^2</a> </span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="square_1.html#K3">^2</a> </span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="topreal9.html#E1:50"><i><font color="Green">E42</font></i></a>, <a class="ref" href="toprns_1.html#D6">TOPRNS_1:def 6</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">l</font></span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p3">(<span class="default"><font color="Maroon">r</font> <a href="square_1.html#K3">^2</a> </span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default">2 <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <span class="p4">(<span class="default"><a href="rvsum_1.html#K14">mlt</a> <font color="Maroon">W1</font>,<font color="Maroon">W2</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">l</font> <a href="square_1.html#K5">^2</a> </span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="square_1.html#K3">^2</a> </span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="rvsum_1.html#T117">RVSUM_1:117</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">l</font></span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p3">(<span class="default"><font color="Maroon">r</font> <a href="square_1.html#K3">^2</a> </span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default">2 <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">l</font></span>)</span> <a href="rvsum_1.html#K10">*</a> <span class="p5">(<span class="default"><a href="rvsum_1.html#K14">mlt</a> <span class="p0">(<span class="default"><font color="Maroon">yf</font> <a href="rvsum_1.html#K8">-</a> <font color="Maroon">xf</font></span>)</span>,<span class="p0">(<span class="default"><font color="Maroon">l</font> <a href="rvsum_1.html#K10">*</a> <span class="p1">(<span class="default"><font color="Maroon">zf</font> <a href="rvsum_1.html#K8">-</a> <font color="Maroon">xf</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">l</font> <a href="square_1.html#K5">^2</a> </span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="square_1.html#K3">^2</a> </span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="topreal9.html#E2:50"><i><font color="Green">E44</font></i></a>, <a class="txt" href="topreal9.html#E1:50_1_2"><i><font color="Green">E46</font></i></a>, <a class="ref" href="rvsum_1.html#T94">RVSUM_1:94</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">l</font></span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p3">(<span class="default"><font color="Maroon">r</font> <a href="square_1.html#K3">^2</a> </span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default">2 <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">l</font></span>)</span> <a href="rvsum_1.html#K10">*</a> <span class="p5">(<span class="default"><font color="Maroon">l</font> <a href="rvsum_1.html#K10">*</a> <span class="p0">(<span class="default"><a href="rvsum_1.html#K14">mlt</a> <span class="p1">(<span class="default"><font color="Maroon">yf</font> <a href="rvsum_1.html#K8">-</a> <font color="Maroon">xf</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">zf</font> <a href="rvsum_1.html#K8">-</a> <font color="Maroon">xf</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">l</font> <a href="square_1.html#K5">^2</a> </span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="square_1.html#K3">^2</a> </span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="rvsum_1.html#T94">RVSUM_1:94</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">l</font></span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p3">(<span class="default"><font color="Maroon">r</font> <a href="square_1.html#K3">^2</a> </span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default">2 <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">l</font></span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">l</font></span>)</span> <a href="rvsum_1.html#K10">*</a> <span class="p5">(<span class="default"><a href="rvsum_1.html#K14">mlt</a> <span class="p0">(<span class="default"><font color="Maroon">yf</font> <a href="rvsum_1.html#K8">-</a> <font color="Maroon">xf</font></span>)</span>,<span class="p0">(<span class="default"><font color="Maroon">zf</font> <a href="rvsum_1.html#K8">-</a> <font color="Maroon">xf</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">l</font> <a href="square_1.html#K5">^2</a> </span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="square_1.html#K3">^2</a> </span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="rvsum_1.html#T71">RVSUM_1:71</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">l</font></span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p3">(<span class="default"><font color="Maroon">r</font> <a href="square_1.html#K3">^2</a> </span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default">2 <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">l</font></span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">l</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p4">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <span class="p5">(<span class="default"><a href="rvsum_1.html#K14">mlt</a> <span class="p0">(<span class="default"><font color="Maroon">yf</font> <a href="rvsum_1.html#K8">-</a> <font color="Maroon">xf</font></span>)</span>,<span class="p0">(<span class="default"><font color="Maroon">zf</font> <a href="rvsum_1.html#K8">-</a> <font color="Maroon">xf</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">l</font> <a href="square_1.html#K5">^2</a> </span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="square_1.html#K3">^2</a> </span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="rvsum_1.html#T117">RVSUM_1:117</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">l</font></span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p3">(<span class="default"><font color="Maroon">r</font> <a href="square_1.html#K3">^2</a> </span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="real_1.html#K4">*</a> <span class="p5">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">l</font></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">l</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <span class="p4">(<span class="default"><a href="rvsum_1.html#K14">mlt</a> <span class="p5">(<span class="default"><font color="Maroon">yf</font> <a href="rvsum_1.html#K8">-</a> <font color="Maroon">xf</font></span>)</span>,<span class="p5">(<span class="default"><font color="Maroon">zf</font> <a href="rvsum_1.html#K8">-</a> <font color="Maroon">xf</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">l</font> <a href="square_1.html#K5">^2</a> </span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="square_1.html#K3">^2</a> </span>)</span></span>)</span>

<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">l</font></span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p3">(<span class="default"><font color="Maroon">r</font> <a href="square_1.html#K3">^2</a> </span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="real_1.html#K4">*</a> <span class="p5">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">l</font></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">l</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p3"><a href="euclid_2.html#K1">|(</a><span class="default"><font color="Maroon">y1</font>,<font color="Maroon">z1</font></span><a href="euclid_2.html#K1">)|</a></span></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">l</font> <a href="square_1.html#K5">^2</a> </span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="square_1.html#K3">^2</a> </span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="topreal9.html#T3" target="_self">Th3</a>, <a class="ref" href="topreal9.html#T4" target="_self">Th4</a>, <a class="ref" href="euclid_2.html#D1">EUCLID_2:def 1</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">l</font></span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p3">(<span class="default"><font color="Maroon">r</font> <a href="square_1.html#K3">^2</a> </span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="real_1.html#K4">*</a> <span class="p5">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">l</font></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">l</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p3"><a href="euclid_2.html#K2">|(</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon">y</font> <a href="euclid.html#K20">-</a> <font color="Maroon">x</font></span>)</span>,<span class="p4">(<span class="default"><font color="Maroon">z</font> <a href="euclid.html#K20">-</a> <font color="Maroon">x</font></span>)</span></span><a href="euclid_2.html#K2">)|</a></span></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">l</font> <a href="square_1.html#K5">^2</a> </span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="square_1.html#K3">^2</a> </span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="euclid_2.html#D2">EUCLID_2:def 2</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default">1 <a href="real_1.html#K5">-</a> <span class="p5">(<span class="default">2 <a href="real_1.html#K4">*</a> <font color="Maroon">l</font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p4">(<span class="default"><font color="Maroon">l</font> <a href="square_1.html#K5">^2</a> </span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p3">(<span class="default"><font color="Maroon">l</font> <a href="square_1.html#K5">^2</a> </span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="square_1.html#K3">^2</a> </span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="real_1.html#K4">*</a> <font color="Maroon">l</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">l</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p2"><a href="euclid_2.html#K2">|(</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">y</font> <a href="euclid.html#K20">-</a> <font color="Maroon">x</font></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon">z</font> <a href="euclid.html#K20">-</a> <font color="Maroon">x</font></span>)</span></span><a href="euclid_2.html#K2">)|</a></span></span>)</span>

;<br/><a NAME="E25:50_1_2"/><b>then </b><i><font color="Green">E122</font></i>: 
<span class="p1">(<span class="default"><span class="p2"><a href="toprns_1.html#K5">|.</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">R</font> <a href="euclid.html#K20">-</a> <font color="Maroon">x</font></span>)</span></span><a href="toprns_1.html#K5">.|</a></span> <a href="square_1.html#K5">^2</a> </span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="square_1.html#K3">^2</a> </span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="real_1.html#K4">*</a> <font color="Maroon">l</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">l</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <span class="p3">(<span class="default"><font color="Maroon">r</font> <a href="square_1.html#K3">^2</a> </span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p2"><a href="euclid_2.html#K2">|(</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">y</font> <a href="euclid.html#K20">-</a> <font color="Maroon">x</font></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon">z</font> <a href="euclid.html#K20">-</a> <font color="Maroon">x</font></span>)</span></span><a href="euclid_2.html#K2">)|</a></span></span>)</span>
 ;<br/><a NAME="E26:50_1_2"/>
 <a href="topreal9.html#K3" target="_self">Sphere</a> <font color="Maroon">x</font>,<font color="Maroon">r</font> <a href="tarski.html#R1">c=</a>  <a href="topreal9.html#K2" target="_self">cl_Ball</a> <font color="Maroon">x</font>,<font color="Maroon">r</font>
 <b>by </b><i/>;<br/><a NAME="E27:50_1_2"/><b>then </b>
 <a href="topreal1.html#K3">LSeg</a> <font color="Maroon">y</font>,<font color="Maroon">z</font> <a href="tarski.html#R1">c=</a>  <a href="topreal9.html#K2" target="_self">cl_Ball</a> <font color="Maroon">x</font>,<font color="Maroon">r</font>
 <b>by </b><i>, , <a class="ref" href="jordan1.html#D1">JORDAN1:def 1</a></i>;<br/><a NAME="E28:50_1_2"/><b>then </b><i><font color="Green">E123</font></i>: 
<span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">R</font> <a href="euclid.html#K20">-</a> <font color="Maroon">x</font></span>)</span></span><a href="toprns_1.html#K5">.|</a></span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">r</font>
 <b>by </b><i>, </i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="50_1_2_3"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:50_1_2_3"/>
<font color="Maroon">l</font> <a href="hidden.html#R1">=</a> 0
 ;<br/><a NAME="E2:50_1_2_3"/><b>then </b>
<font color="Maroon">R</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font>
 <b>by </b><i>, </i>;<br/><a NAME="E3:50_1_2_3"/><b>then </b>
<font color="Maroon">R</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">z</font></span><a href="tarski.html#K2">}</a></span>
 <b>by </b><i><a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/><b>hence </b><a NAME="E4:50_1_2_3"/>
contradiction
 <b>by </b><i>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/></div><b>end;</b></div><a NAME="E30:50_1_2"/><b>then </b><i><font color="Green">E124</font></i>: 
2 <a href="real_1.html#K4">*</a> <font color="Maroon">l</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 <b>by </b><i>, <a class="ref" href="xreal_1.html#T131">XREAL_1:131</a></i>;<br/><a NAME="E31:50_1_2"/><i><font color="Green">E125</font></i>: 
1 <a href="real_1.html#K5">-</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> 1 <a href="real_1.html#K5">-</a> <font color="Maroon">l</font>
 <b>by </b><i>, <a class="ref" href="real_1.html#T92">REAL_1:92</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="50_1_2_4"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:50_1_2_4"/>
1 <a href="real_1.html#K5">-</a> <font color="Maroon">l</font> <a href="hidden.html#R1">=</a> 0
 ;<br/><a NAME="E2:50_1_2_4"/><b>then </b>
<span class="p1">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">l</font></span>)</span> <a href="real_1.html#K3">+</a> <font color="Maroon">l</font> <a href="hidden.html#R1">=</a> 0 <a href="real_1.html#K3">+</a> <font color="Maroon">l</font>
 ;<br/><a NAME="E3:50_1_2_4"/><b>then </b>
<font color="Maroon">l</font> <a href="hidden.html#R1">=</a> 1
 ;<br/><a NAME="E4:50_1_2_4"/><b>then </b>
<font color="Maroon">R</font> <a href="hidden.html#R1">=</a> <font color="Maroon">z</font>
 <b>by </b><i>, </i>;<br/><a NAME="E5:50_1_2_4"/><b>then </b>
<font color="Maroon">R</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">z</font></span><a href="tarski.html#K2">}</a></span>
 <b>by </b><i><a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/><b>hence </b><a NAME="E6:50_1_2_4"/>
contradiction
 <b>by </b><i>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/></div><b>end;</b></div><a NAME="E33:50_1_2"/><b>then </b><i><font color="Green">E126</font></i>: 
<span class="p1">(<span class="default">2 <a href="real_1.html#K4">*</a> <font color="Maroon">l</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">l</font></span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 <b>by </b><i>, , <a class="ref" href="xreal_1.html#T131">XREAL_1:131</a></i>;<br/><a NAME="E34:50_1_2"/><i><font color="Green">E127</font></i>: 
 <a href="rvsum_1.html#K15">Sum</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K12">sqr</a> <span class="p2">(<span class="default"><font color="Maroon">yf</font> <a href="rvsum_1.html#K8">-</a> <font color="Maroon">zf</font></span>)</span></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 <b>by </b><i><a class="ref" href="rvsum_1.html#T116">RVSUM_1:116</a></i>;<br/><a NAME="E35:50_1_2"/><i><font color="Green">E128</font></i>: 
<font color="Maroon">y</font> <a href="euclid.html#K20">-</a> <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">yf</font> <a href="rvsum_1.html#K8">-</a> <font color="Maroon">zf</font>
 <b>by </b><i><a class="ref" href="euclid.html#D13">EUCLID:def 13</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="50_1_2_5"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:50_1_2_5"/>
<span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">R</font> <a href="euclid.html#K20">-</a> <font color="Maroon">x</font></span>)</span></span><a href="toprns_1.html#K5">.|</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">r</font>
 ;<br/><a NAME="E2:50_1_2_5"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="real_1.html#K4">*</a> <font color="Maroon">l</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">l</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <span class="p3">(<span class="default"><font color="Maroon">r</font> <a href="square_1.html#K3">^2</a> </span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p2"><a href="euclid_2.html#K2">|(</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">y</font> <a href="euclid.html#K20">-</a> <font color="Maroon">x</font></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon">z</font> <a href="euclid.html#K20">-</a> <font color="Maroon">x</font></span>)</span></span><a href="euclid_2.html#K2">)|</a></span></span>)</span> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i/>;<br/><a NAME="E3:50_1_2_5"/><b>then </b>
<span class="p1"><a href="euclid_2.html#K2">|(</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">y</font> <a href="euclid.html#K20">-</a> <font color="Maroon">x</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">z</font> <a href="euclid.html#K20">-</a> <font color="Maroon">x</font></span>)</span></span><a href="euclid_2.html#K2">)|</a></span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="square_1.html#K3">^2</a> </span>)</span> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i>, <a class="ref" href="xcmplx_1.html#T6">XCMPLX_1:6</a></i>;<br/><a NAME="E4:50_1_2_5"/><b>then </b>
<span class="p1"><a href="euclid_2.html#K2">|(</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">y</font> <a href="euclid.html#K20">-</a> <font color="Maroon">x</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">z</font> <a href="euclid.html#K20">-</a> <font color="Maroon">x</font></span>)</span></span><a href="euclid_2.html#K2">)|</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">r</font> <a href="square_1.html#K3">^2</a> 
 ;<br/><b>then </b>0 = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="toprns_1.html#K5">|.</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon">y</font> <a href="euclid.html#K20">-</a> <font color="Maroon">x</font></span>)</span></span><a href="toprns_1.html#K5">.|</a></span> <a href="square_1.html#K5">^2</a> </span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p2">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <span class="p3"><a href="euclid_2.html#K2">|(</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon">y</font> <a href="euclid.html#K20">-</a> <font color="Maroon">x</font></span>)</span>,<span class="p4">(<span class="default"><font color="Maroon">z</font> <a href="euclid.html#K20">-</a> <font color="Maroon">x</font></span>)</span></span><a href="euclid_2.html#K2">)|</a></span></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><span class="p2"><a href="toprns_1.html#K5">|.</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">z</font> <a href="euclid.html#K20">-</a> <font color="Maroon">x</font></span>)</span></span><a href="toprns_1.html#K5">.|</a></span> <a href="square_1.html#K5">^2</a> </span>)</span>
<b>by </b><i><a class="ref" href="topreal9.html#T2" target="_self">Th2</a>, <a class="txt" href="topreal9.html#E1:50"><i><font color="Green">E42</font></i></a></i>
<br/>.= 
<span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">y</font> <a href="euclid.html#K20">-</a> <font color="Maroon">x</font></span>)</span> <a href="euclid.html#K20">-</a> <span class="p3">(<span class="default"><font color="Maroon">z</font> <a href="euclid.html#K20">-</a> <font color="Maroon">x</font></span>)</span></span>)</span></span><a href="toprns_1.html#K5">.|</a></span> <a href="square_1.html#K5">^2</a> 
<b>by </b><i><a class="ref" href="euclid_2.html#T68">EUCLID_2:68</a></i>
<br/>.= 
<span class="p1"><a href="toprns_1.html#K5">|.</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">y</font> <a href="euclid.html#K20">-</a> <font color="Maroon">x</font></span>)</span> <a href="euclid.html#K20">-</a> <font color="Maroon">z</font></span>)</span> <a href="euclid.html#K17">+</a> <font color="Maroon">x</font></span>)</span></span><a href="toprns_1.html#K5">.|</a></span> <a href="square_1.html#K5">^2</a> 
<b>by </b><i><a class="ref" href="euclid.html#T51">EUCLID:51</a></i>
<br/>.= 
<span class="p1"><a href="toprns_1.html#K5">|.</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">y</font> <a href="euclid.html#K20">-</a> <font color="Maroon">x</font></span>)</span> <a href="euclid.html#K17">+</a> <font color="Maroon">x</font></span>)</span> <a href="euclid.html#K20">-</a> <font color="Maroon">z</font></span>)</span></span><a href="toprns_1.html#K5">.|</a></span> <a href="square_1.html#K5">^2</a> 
<b>by </b><i><a class="ref" href="jordan2c.html#T9">JORDAN2C:9</a></i>
<br/>.= 
<span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">y</font> <a href="euclid.html#K20">-</a> <font color="Maroon">z</font></span>)</span></span><a href="toprns_1.html#K5">.|</a></span> <a href="square_1.html#K5">^2</a> 
<b>by </b><i><a class="ref" href="euclid.html#T52">EUCLID:52</a></i>
<br/>.= 
<span class="p1"><a href="euclid.html#K12">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">yf</font> <a href="rvsum_1.html#K8">-</a> <font color="Maroon">zf</font></span>)</span></span><a href="euclid.html#K12">.|</a></span> <a href="square_1.html#K5">^2</a> 
<b>by </b><i>, <a class="ref" href="toprns_1.html#D6">TOPRNS_1:def 6</a></i>
<br/>.= 
 <a href="rvsum_1.html#K15">Sum</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K12">sqr</a> <span class="p2">(<span class="default"><font color="Maroon">yf</font> <a href="rvsum_1.html#K8">-</a> <font color="Maroon">zf</font></span>)</span></span>)</span>
<b>by </b><i>, <a class="ref" href="square_1.html#D4">SQUARE_1:def 4</a></i>
;<br/><a NAME="E6:50_1_2_5"/><b>then </b>
<font color="Maroon">yf</font> <a href="rvsum_1.html#K8">-</a> <font color="Maroon">zf</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> <a href="finseq_2.html#K2">|-&gt;</a> 0
 <b>by </b><i><a class="ref" href="rvsum_1.html#T121">RVSUM_1:121</a></i>;<br/><b>hence </b><a NAME="E7:50_1_2_5"/>
contradiction
 <b>by </b><i>, <a class="ref" href="rvsum_1.html#T59">RVSUM_1:59</a></i>;<br/></div><b>end;</b></div><a NAME="E37:50_1_2"/><b>then </b>
<span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">R</font> <a href="euclid.html#K20">-</a> <font color="Maroon">x</font></span>)</span></span><a href="toprns_1.html#K5">.|</a></span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r</font>
 <b>by </b><i><a class="ref" href="topreal9.html#T6" target="_self">Th6</a>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/><b>hence </b><a NAME="E38:50_1_2"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="topreal9.html#K1" target="_self">Ball</a> <font color="Maroon">x</font>,<font color="Maroon">r</font>
 ;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div>
