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<b>let </b><font color="Maroon" title="c1">n</font> be    <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="c2">a</font>, <font color="Maroon" title="c3">b</font>, <font color="Maroon" title="c4">r</font> be  <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> ;<br/><b>let </b><font color="Maroon" title="c5">x</font>, <font color="Maroon" title="c6">z</font>, <font color="Maroon" title="c7">y</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="c1">n</font></span>)</span>;<br/>













<b>assume </b><b>that </b><br/><a NAME="E1:29"/><i><font color="Green" title="E18">A1</font></i>: 
<font color="Maroon" title="c2">a</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c3">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <b>and </b><br/><a NAME="E2:29"/><i><font color="Green" title="E19">A2</font></i>: 
<span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <font color="Maroon" title="c2">a</font></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <font color="Maroon" title="c3">b</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <b>and </b><br/><a NAME="E3:29"/><i><font color="Green" title="E20">A3</font></i>: 
<font color="Maroon" title="c3">b</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 0
 <b>and </b><br/><a NAME="E4:29"/><i><font color="Green" title="E21">A4</font></i>: 
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal9.html#K2" title="TOPREAL9:func.2">cl_Ball</a> <font color="Maroon" title="c6">z</font>,<font color="Maroon" title="c4">r</font>
 <b>and </b><br/><a NAME="E5:29"/><i><font color="Green" title="E22">A5</font></i>: 
<font color="Maroon" title="c7">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal9.html#K1" title="TOPREAL9:func.1">Ball</a> <font color="Maroon" title="c6">z</font>,<font color="Maroon" title="c4">r</font>
 ;<br/>

<b>reconsider </b><font color="Maroon" title="c8">w</font> = <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="c5">x</font></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">b</font> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <font color="Maroon" title="c7">y</font></span>)</span> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K14" title="EUCLID:func.14">Euclid</a> <font color="Maroon" title="c1">n</font></span>)</span> <b>by </b><i><a class="ref" href="topreal3.html#T13" title="TOPREAL3:th.13">TOPREAL3:13</a></i>;<br/>
<i><font color="Green" title="E23">A6</font></i>: <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">z</font></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">b</font> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <font color="Maroon" title="c6">z</font></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c3">b</font></span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <font color="Maroon" title="c6">z</font>
<b>by </b><i><a class="ref" href="euclid.html#T37" title="EUCLID:th.37">EUCLID:37</a></i>
<br/>.= 
<font color="Maroon" title="c6">z</font>
<b>by </b><i><a class="txt" href="topreal9.html#E1:29"><i><font color="Green" title="E18">A1</font></i></a>, <a class="ref" href="euclid.html#T33" title="EUCLID:th.33">EUCLID:33</a></i>
;<br/>
<i><font color="Green" title="E24">A7</font></i>: <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="c2">a</font> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <font color="Maroon" title="c6">z</font></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">+</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">b</font> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <font color="Maroon" title="c6">z</font></span>)</span></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="c2">a</font> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <font color="Maroon" title="c5">x</font></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">+</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">b</font> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <font color="Maroon" title="c7">y</font></span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> = 
<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="c2">a</font> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <font color="Maroon" title="c6">z</font></span>)</span> <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="c2">a</font> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <font color="Maroon" title="c5">x</font></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">+</a> <span class="p5">(<span class="default"><font color="Maroon" title="c3">b</font> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <font color="Maroon" title="c7">y</font></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">b</font> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <font color="Maroon" title="c6">z</font></span>)</span></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span>
<b>by </b><i><a class="ref" href="jordan2c.html#T9" title="JORDAN2C:th.9">JORDAN2C: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"><font color="Maroon" title="c2">a</font> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <font color="Maroon" title="c6">z</font></span>)</span> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <span class="p5">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <font color="Maroon" title="c5">x</font></span>)</span></span>)</span> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">b</font> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <font color="Maroon" title="c7">y</font></span>)</span></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">b</font> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <font color="Maroon" title="c6">z</font></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#T50" title="EUCLID:th.50">EUCLID:50</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"><font color="Maroon" title="c2">a</font> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <font color="Maroon" title="c6">z</font></span>)</span> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <span class="p5">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <font color="Maroon" title="c5">x</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="c3">b</font> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <font color="Maroon" title="c6">z</font></span>)</span></span>)</span> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">b</font> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <font color="Maroon" title="c7">y</font></span>)</span></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span>
<b>by </b><i><a class="ref" href="jordan2c.html#T9" title="JORDAN2C:th.9">JORDAN2C: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"><font color="Maroon" title="c2">a</font> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <font color="Maroon" title="c6">z</font></span>)</span> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <span class="p4">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <font color="Maroon" title="c5">x</font></span>)</span></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">b</font> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <font color="Maroon" title="c6">z</font></span>)</span> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">b</font> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <font color="Maroon" title="c7">y</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#T49" title="EUCLID:th.49">EUCLID:49</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="c2">a</font> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c6">z</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c5">x</font></span>)</span></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">b</font> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <font color="Maroon" title="c6">z</font></span>)</span> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">b</font> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <font color="Maroon" title="c7">y</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="c2">a</font> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c6">z</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c5">x</font></span>)</span></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">b</font> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c6">z</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c7">y</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/>
<a NAME="E9:29"/><i><font color="Green" title="E25">A8</font></i>: 
<font color="Maroon" title="c2">a</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/>
<a NAME="E10:29"/><i><font color="Green" title="E26">A9</font></i>: 
<font color="Maroon" title="c3">b</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/>
<a NAME="E11:29"/><i><font color="Green" title="E27">A10</font></i>: 
0 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <font color="Maroon" title="c2">a</font>
 
<b>by </b><i><a class="ref" href="complex1.html#T132" title="COMPLEX1:th.132">COMPLEX1:132</a></i>;<br/>
<a NAME="E12:29"/><i><font color="Green" title="E28">A11</font></i>: 
0 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a>  <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <font color="Maroon" title="c3">b</font>
 
<b>by </b><i><a class="txt" href="topreal9.html#E3:29"><i><font color="Green" title="E20">A3</font></i></a>, <a class="ref" href="complex1.html#T133" title="COMPLEX1:th.133">COMPLEX1:133</a></i>;<br/>
<a NAME="E13:29"/>
( <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">x</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c6">z</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="c4">r</font> &amp; <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="c7">y</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c6">z</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="c4">r</font> )
 
<b>by </b><i><a class="txt" href="topreal9.html#E4:29"><i><font color="Green" title="E21">A4</font></i></a>, <a class="txt" href="topreal9.html#E5:29"><i><font color="Green" title="E22">A5</font></i></a>, <a class="ref" href="topreal9.html#T7" target="_self" title="TOPREAL9:th.7">Th7</a>, <a class="ref" href="topreal9.html#T8" target="_self" title="TOPREAL9:th.8">Th8</a></i>;<br/>
<a NAME="E14:29"/><b>then </b>
( <span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">z</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c5">x</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="c4">r</font> &amp; <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="c6">z</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c7">y</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="c4">r</font> )
 
<b>by </b><i><a class="ref" href="toprns_1.html#T28" title="TOPRNS_1:th.28">TOPRNS_1:28</a></i>;<br/>
<a NAME="E15:29"/><b>then </b>
( <span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <font color="Maroon" title="c2">a</font></span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <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="c6">z</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c5">x</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> <span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <font color="Maroon" title="c2">a</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c4">r</font> &amp; <span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <font color="Maroon" title="c3">b</font></span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <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="c6">z</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c7">y</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> <span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <font color="Maroon" title="c3">b</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c4">r</font> )
 
<b>by </b><i><a class="txt" href="topreal9.html#E11:29"><i><font color="Green" title="E27">A10</font></i></a>, <a class="txt" href="topreal9.html#E12:29"><i><font color="Green" title="E28">A11</font></i></a>, <a class="ref" href="xreal_1.html#T66" title="XREAL_1:th.66">XREAL_1:66</a>, <a class="ref" href="xreal_1.html#T70" title="XREAL_1:th.70">XREAL_1:70</a></i>;<br/>
<a NAME="E16:29"/><b>then </b>
<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="c2">a</font></span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</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">z</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c5">x</font></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</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="c3">b</font></span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</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">z</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c7">y</font></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</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="c2">a</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c4">r</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <font color="Maroon" title="c3">b</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c4">r</font></span>)</span>
 
<b>by </b><i><a class="ref" href="xreal_1.html#T10" title="XREAL_1:th.10">XREAL_1:10</a></i>;<br/>
<a NAME="E17:29"/><b>then </b>
<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="c2">a</font></span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</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">z</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c5">x</font></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</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="c3">b</font></span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</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">z</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c7">y</font></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</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="c2">a</font></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p2">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <font color="Maroon" title="c3">b</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c4">r</font>
 
;<br/>
<a NAME="E18:29"/><b>then </b>
<span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c6">z</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c5">x</font></span>)</span></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="real_1.html#K3" title="REAL_1:func.3">+</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="c3">b</font></span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</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">z</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c7">y</font></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> 1 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c4">r</font>
 
<b>by </b><i><a class="txt" href="topreal9.html#E2:29"><i><font color="Green" title="E19">A2</font></i></a>, <a class="txt" href="topreal9.html#E9:29"><i><font color="Green" title="E25">A8</font></i></a>, <a class="ref" href="toprns_1.html#T8" title="TOPRNS_1:th.8">TOPRNS_1:8</a></i>;<br/>
<a NAME="E19:29"/><b>then </b><i><font color="Green" title="E29">A12</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="c2">a</font> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c6">z</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c5">x</font></span>)</span></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <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="c3">b</font> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c6">z</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c7">y</font></span>)</span></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="c4">r</font>
 
<b>by </b><i><a class="txt" href="topreal9.html#E10:29"><i><font color="Green" title="E26">A9</font></i></a>, <a class="ref" href="toprns_1.html#T8" title="TOPRNS_1:th.8">TOPRNS_1:8</a></i>;<br/>
<a NAME="E20:29"/>
<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="c2">a</font> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <font color="Maroon" title="c6">z</font></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">+</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">b</font> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <font color="Maroon" title="c6">z</font></span>)</span></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="c2">a</font> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <font color="Maroon" title="c5">x</font></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">+</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">b</font> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <font color="Maroon" title="c7">y</font></span>)</span></span>)</span></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> <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="c2">a</font> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c6">z</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c5">x</font></span>)</span></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <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="c3">b</font> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c6">z</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c7">y</font></span>)</span></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span>
 
<b>by </b><i><a class="txt" href="topreal9.html#E8:29"><i><font color="Green" title="E24">A7</font></i></a>, <a class="ref" href="toprns_1.html#T30" title="TOPRNS_1:th.30">TOPRNS_1:30</a></i>;<br/>
<a NAME="E21:29"/><b>then </b>
<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="c2">a</font> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <font color="Maroon" title="c6">z</font></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">+</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">b</font> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <font color="Maroon" title="c6">z</font></span>)</span></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="c2">a</font> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <font color="Maroon" title="c5">x</font></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">+</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">b</font> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <font color="Maroon" title="c7">y</font></span>)</span></span>)</span></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="c4">r</font>
 
<b>by </b><i><a class="txt" href="topreal9.html#E19:29"><i><font color="Green" title="E29">A12</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<a NAME="E22:29"/><b>then </b>
<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="c2">a</font> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <font color="Maroon" title="c5">x</font></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">+</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">b</font> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <font color="Maroon" title="c7">y</font></span>)</span></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="c2">a</font> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <font color="Maroon" title="c6">z</font></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">+</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">b</font> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <font color="Maroon" title="c6">z</font></span>)</span></span>)</span></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="c4">r</font>
 
<b>by </b><i><a class="ref" href="toprns_1.html#T28" title="TOPRNS_1:th.28">TOPRNS_1:28</a></i>;<br/>
<b>hence </b><a NAME="E23:29"/>
<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="c5">x</font></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">b</font> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <font color="Maroon" title="c7">y</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal9.html#K1" title="TOPREAL9:func.1">Ball</a> <font color="Maroon" title="c6">z</font>,<font color="Maroon" title="c4">r</font>
 <b>by </b><i><a class="txt" href="topreal9.html#E7:29"><i><font color="Green" title="E23">A6</font></i></a></i>;<br/>


</div>
