<?xml version="1.0"?>
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<span class="kw">let </span><font color="Maroon" title="c1">x</font>, <font color="Maroon" title="c2">y</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> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">x</font> &amp; 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">y</font> &amp;  <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <font color="Maroon" title="c2">y</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <font color="Maroon" title="c1">x</font> implies  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">y</font> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <span class="p2">(<span class="default"><a href="square_1.html#K7" title="SQUARE_1:func.7">sqrt</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="square_1.html#K3" title="SQUARE_1:func.3">^2</a></span>)</span> <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> 1</span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <span class="p2">(<span class="default"><a href="square_1.html#K7" title="SQUARE_1:func.7">sqrt</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c2">y</font> <a href="square_1.html#K3" title="SQUARE_1:func.3">^2</a></span>)</span> <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> 1</span>)</span></span>)</span></span>)</span> )</span><br/>



<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:29"/><span class="lab"><font color="Green" title="E24">A1</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">x</font>
 <span class="kw">and </span><br/><a NAME="E2:29"/><span class="lab"><font color="Green" title="E25">A2</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">y</font>
 <span class="kw">and </span><br/><a NAME="E3:29"/><span class="lab"><font color="Green" title="E26">A3</font></span>: 
 <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <font color="Maroon" title="c2">y</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <font color="Maroon" title="c1">x</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">y</font> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <span class="p2">(<span class="default"><a href="square_1.html#K7" title="SQUARE_1:func.7">sqrt</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="square_1.html#K3" title="SQUARE_1:func.3">^2</a></span>)</span> <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> 1</span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <span class="p2">(<span class="default"><a href="square_1.html#K7" title="SQUARE_1:func.7">sqrt</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c2">y</font> <a href="square_1.html#K3" title="SQUARE_1:func.3">^2</a></span>)</span> <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> 1</span>)</span></span>)</span></span>)</span></span><br/>

<a NAME="E4:29"/><span class="lab"><font color="Green" title="E27">A4</font></span>: 
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">y</font> <a href="square_1.html#K3" title="SQUARE_1:func.3">^2</a></span>)</span> <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:29"><span class="lab"><font color="Green" title="E25">A2</font></span></a>, <a class="txt" href="sin_cos7.html#E8"><span class="lab"><font color="Green" title="E6">Lm3</font></span></a></span>;<br/>
<a NAME="E5:29"/><span class="lab"><font color="Green" title="E28">A5</font></span>: 
( <span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <font color="Maroon" title="c1">x</font></span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">x</font> <a href="square_1.html#K3" title="SQUARE_1:func.3">^2</a>  &amp; <span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <font color="Maroon" title="c2">y</font></span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">y</font> <a href="square_1.html#K3" title="SQUARE_1:func.3">^2</a>  )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="complex1.html#T75" title="COMPLEX1:th.75">COMPLEX1:75</a></span>;<br/>
<a NAME="E6:29"/>
1 <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">y</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:29"><span class="lab"><font color="Green" title="E25">A2</font></span></a>, <a class="ref" href="absvalue.html#D1" title="ABSVALUE:def.1">ABSVALUE:def 1</a></span>;<br/>
<a NAME="E7:29"/><span class="kw">then </span>
<font color="Maroon" title="c2">y</font> <a href="square_1.html#K3" title="SQUARE_1:func.3">^2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">x</font> <a href="square_1.html#K3" title="SQUARE_1:func.3">^2</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:29"><span class="lab"><font color="Green" title="E26">A3</font></span></a>, <a class="txt" href="#E5:29"><span class="lab"><font color="Green" title="E28">A5</font></span></a>, <a class="ref" href="square_1.html#T15" title="SQUARE_1:th.15">SQUARE_1:15</a></span>;<br/>
<a NAME="E8:29"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="binop_2.html#K19" title="BINOP_2:func.19">-</a> 1</span>)</span> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="square_1.html#K3" title="SQUARE_1:func.3">^2</a></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="binop_2.html#K19" title="BINOP_2:func.19">-</a> 1</span>)</span> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">y</font> <a href="square_1.html#K3" title="SQUARE_1:func.3">^2</a></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T65" title="XREAL_1:th.65">XREAL_1:65</a></span>;<br/>
<a NAME="E9:29"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="binop_2.html#K7" title="BINOP_2:func.7">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="square_1.html#K3" title="SQUARE_1:func.3">^2</a></span>)</span></span>)</span> <a href="binop_2.html#K9" title="BINOP_2:func.9">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="square_1.html#K3" title="SQUARE_1:func.3">^2</a></span>)</span> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">y</font> <a href="square_1.html#K3" title="SQUARE_1:func.3">^2</a></span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="binop_2.html#K7" title="BINOP_2:func.7">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">y</font> <a href="square_1.html#K3" title="SQUARE_1:func.3">^2</a></span>)</span></span>)</span> <a href="binop_2.html#K9" title="BINOP_2:func.9">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="square_1.html#K3" title="SQUARE_1:func.3">^2</a></span>)</span> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">y</font> <a href="square_1.html#K3" title="SQUARE_1:func.3">^2</a></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T6" title="XREAL_1:th.6">XREAL_1:6</a></span>;<br/>
<a NAME="E10:29"/><span class="kw">then </span>
(  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">x</font> <a href="square_1.html#K3" title="SQUARE_1:func.3">^2</a>  &amp;  <a href="square_1.html#K7" title="SQUARE_1:func.7">sqrt</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="square_1.html#K3" title="SQUARE_1:func.3">^2</a></span>)</span> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">y</font> <a href="square_1.html#K3" title="SQUARE_1:func.3">^2</a></span>)</span> <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> 1</span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="square_1.html#K7" title="SQUARE_1:func.7">sqrt</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">y</font> <a href="square_1.html#K3" title="SQUARE_1:func.3">^2</a></span>)</span> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="square_1.html#K3" title="SQUARE_1:func.3">^2</a></span>)</span> <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> 1</span>)</span></span>)</span> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:29"><span class="lab"><font color="Green" title="E24">A1</font></span></a>, <a class="txt" href="#E4:29"><span class="lab"><font color="Green" title="E27">A4</font></span></a>, <a class="ref" href="square_1.html#T26" title="SQUARE_1:th.26">SQUARE_1:26</a></span>;<br/>
<a NAME="E11:29"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="square_1.html#K6" title="SQUARE_1:func.6">sqrt</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="square_1.html#K3" title="SQUARE_1:func.3">^2</a></span>)</span></span>)</span> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <span class="p1">(<span class="default"><a href="square_1.html#K7" title="SQUARE_1:func.7">sqrt</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">y</font> <a href="square_1.html#K3" title="SQUARE_1:func.3">^2</a></span>)</span> <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> 1</span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="square_1.html#K7" title="SQUARE_1:func.7">sqrt</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">y</font> <a href="square_1.html#K3" title="SQUARE_1:func.3">^2</a></span>)</span> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="square_1.html#K3" title="SQUARE_1:func.3">^2</a></span>)</span> <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> 1</span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:29"><span class="lab"><font color="Green" title="E27">A4</font></span></a>, <a class="ref" href="square_1.html#T29" title="SQUARE_1:th.29">SQUARE_1:29</a></span>;<br/>
<a NAME="E12:29"/><span class="kw">then </span><span class="lab"><font color="Green" title="E29">A6</font></span>: 
<font color="Maroon" title="c1">x</font> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <span class="p1">(<span class="default"><a href="square_1.html#K7" title="SQUARE_1:func.7">sqrt</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">y</font> <a href="square_1.html#K3" title="SQUARE_1:func.3">^2</a></span>)</span> <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> 1</span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="square_1.html#K7" title="SQUARE_1:func.7">sqrt</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">y</font> <a href="square_1.html#K3" title="SQUARE_1:func.3">^2</a></span>)</span> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="square_1.html#K3" title="SQUARE_1:func.3">^2</a></span>)</span> <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> 1</span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:29"><span class="lab"><font color="Green" title="E24">A1</font></span></a>, <a class="ref" href="square_1.html#T22" title="SQUARE_1:th.22">SQUARE_1:22</a></span>;<br/>
<a NAME="E13:29"/>
(  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">y</font> <a href="square_1.html#K3" title="SQUARE_1:func.3">^2</a>  &amp;  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="square_1.html#K3" title="SQUARE_1:func.3">^2</a></span>)</span> <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> 1 )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:29"><span class="lab"><font color="Green" title="E24">A1</font></span></a>, <a class="txt" href="sin_cos7.html#E8"><span class="lab"><font color="Green" title="E6">Lm3</font></span></a>, <a class="ref" href="xreal_1.html#T63" title="XREAL_1:th.63">XREAL_1:63</a></span>;<br/>
<a NAME="E14:29"/><span class="kw">then </span>
<font color="Maroon" title="c1">x</font> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <span class="p1">(<span class="default"><a href="square_1.html#K7" title="SQUARE_1:func.7">sqrt</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">y</font> <a href="square_1.html#K3" title="SQUARE_1:func.3">^2</a></span>)</span> <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> 1</span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="square_1.html#K6" title="SQUARE_1:func.6">sqrt</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">y</font> <a href="square_1.html#K3" title="SQUARE_1:func.3">^2</a></span>)</span></span>)</span> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <span class="p1">(<span class="default"><a href="square_1.html#K7" title="SQUARE_1:func.7">sqrt</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="square_1.html#K3" title="SQUARE_1:func.3">^2</a></span>)</span> <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> 1</span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:29"><span class="lab"><font color="Green" title="E29">A6</font></span></a>, <a class="ref" href="square_1.html#T29" title="SQUARE_1:th.29">SQUARE_1:29</a></span>;<br/>
<a NAME="E15:29"/><span class="kw">then </span>
<font color="Maroon" title="c1">x</font> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <span class="p1">(<span class="default"><a href="square_1.html#K7" title="SQUARE_1:func.7">sqrt</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">y</font> <a href="square_1.html#K3" title="SQUARE_1:func.3">^2</a></span>)</span> <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> 1</span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">y</font> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <span class="p1">(<span class="default"><a href="square_1.html#K7" title="SQUARE_1:func.7">sqrt</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="square_1.html#K3" title="SQUARE_1:func.3">^2</a></span>)</span> <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> 1</span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:29"><span class="lab"><font color="Green" title="E25">A2</font></span></a>, <a class="ref" href="square_1.html#T22" title="SQUARE_1:th.22">SQUARE_1:22</a></span>;<br/>
<a NAME="E16:29"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <span class="p2">(<span class="default"><a href="square_1.html#K7" title="SQUARE_1:func.7">sqrt</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c2">y</font> <a href="square_1.html#K3" title="SQUARE_1:func.3">^2</a></span>)</span> <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> 1</span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <span class="p2">(<span class="default"><a href="square_1.html#K7" title="SQUARE_1:func.7">sqrt</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c2">y</font> <a href="square_1.html#K3" title="SQUARE_1:func.3">^2</a></span>)</span> <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> 1</span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">y</font> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <span class="p2">(<span class="default"><a href="square_1.html#K7" title="SQUARE_1:func.7">sqrt</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="square_1.html#K3" title="SQUARE_1:func.3">^2</a></span>)</span> <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> 1</span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <span class="p2">(<span class="default"><a href="square_1.html#K7" title="SQUARE_1:func.7">sqrt</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c2">y</font> <a href="square_1.html#K3" title="SQUARE_1:func.3">^2</a></span>)</span> <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> 1</span>)</span></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T13" title="XREAL_1:th.13">XREAL_1:13</a></span>;<br/>
<span class="kw">hence </span><a NAME="E17:29"/>
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">y</font> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <span class="p2">(<span class="default"><a href="square_1.html#K7" title="SQUARE_1:func.7">sqrt</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="square_1.html#K3" title="SQUARE_1:func.3">^2</a></span>)</span> <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> 1</span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <span class="p2">(<span class="default"><a href="square_1.html#K7" title="SQUARE_1:func.7">sqrt</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c2">y</font> <a href="square_1.html#K3" title="SQUARE_1:func.3">^2</a></span>)</span> <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> 1</span>)</span></span>)</span></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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