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<span class="kw">let </span><font color="Maroon" title="c1">a</font>, <font color="Maroon" title="c2">b</font>, <font color="Maroon" title="c3">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> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c3">r</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  implies (  <a href="topreal9.html#K1" title="TOPREAL9:func.1">Ball</a> (<span class="p1"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span>,<font color="Maroon" title="c3">r</font>) <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="topgen_5.html#K2" title="TOPGEN_5:func.2">y&gt;=0-plane</a>  iff <font color="Maroon" title="c3">r</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">b</font> ) )</span><br/>

<span class="kw">assume </span><a NAME="E1:23"/><span class="lab"><font color="Green" title="E17">A1</font></span>: 
<font color="Maroon" title="c3">r</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  <a href="topreal9.html#K1" title="TOPREAL9:func.1">Ball</a> (<span class="p1"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span>,<font color="Maroon" title="c3">r</font>) <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="topgen_5.html#K2" title="TOPGEN_5:func.2">y&gt;=0-plane</a>  iff <font color="Maroon" title="c3">r</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">b</font> )</span><br/>

<div><span class="kw">hereby </span> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c3">r</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">b</font> implies  <a href="topreal9.html#K1" title="TOPREAL9:func.1">Ball</a> (<span class="p1"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span>,<font color="Maroon" title="c3">r</font>) <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="topgen_5.html#K2" title="TOPGEN_5:func.2">y&gt;=0-plane</a>  )</span>
<div class="add"><a NAME="E1:23_1"/><span class="lab"><font color="Green" title="E18">A2</font></span>: 
<span class="p1"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal9.html#K1" title="TOPREAL9:func.1">Ball</a> (<span class="p1"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span>,<font color="Maroon" title="c3">r</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:23"><span class="lab"><font color="Green" title="E17">A1</font></span></a>, <a class="ref" href="topgen_5.html#T13" target="_self" title="TOPGEN_5:th.13">Th17</a></span>;<br/><span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E2:23_1"/><span class="lab"><font color="Green" title="E19">A3</font></span>: 
 <a href="topreal9.html#K1" title="TOPREAL9:func.1">Ball</a> (<span class="p1"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span>,<font color="Maroon" title="c3">r</font>) <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="topgen_5.html#K2" title="TOPGEN_5:func.2">y&gt;=0-plane</a> 
 <span class="kw">and </span><br/><a NAME="E3:23_1"/><span class="lab"><font color="Green" title="E20">A4</font></span>: 
<font color="Maroon" title="c3">r</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c2">b</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><span class="kw">reconsider </span><font color="Maroon" title="c4">b</font> = <font color="Maroon" title="c2">b</font> as   non  <a href="xxreal_0.html#V3" title="XXREAL_0:attr.3">negative</a>  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:23_1"><span class="lab"><font color="Green" title="E18">A2</font></span></a>, <a class="txt" href="#E2:23_1"><span class="lab"><font color="Green" title="E19">A3</font></span></a>, <a class="ref" href="topgen_5.html#T18" target="_self" title="TOPGEN_5:th.18">Th22</a>, <a class="ref" href="xreal_0.html#D1" title="XREAL_0:def.1">XREAL_0:def 1</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c5">br</font> = <font color="Maroon" title="c4">b</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c3">r</font> as   <a href="xxreal_0.html#V3" title="XXREAL_0:attr.3">negative</a>  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:23_1"><span class="lab"><font color="Green" title="E20">A4</font></span></a>, <a class="ref" href="xreal_1.html#T49" title="XREAL_1:th.49">XREAL_1:49</a></span>;<br/><span class="kw">set </span><font color="Maroon" title="c6">y</font> = <font color="Maroon" title="c5">br</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2;<br/><span class="kw">reconsider </span><font color="Maroon" title="c7">r</font> = <font color="Maroon" title="c3">r</font> as   <a href="xxreal_0.html#V2" title="XXREAL_0:attr.2">positive</a>  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:23"><span class="lab"><font color="Green" title="E17">A1</font></span></a>, <a class="ref" href="xreal_0.html#D1" title="XREAL_0:def.1">XREAL_0:def 1</a></span>;<br/><a NAME="E7:23_1"/>
<span class="p1"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><font color="Maroon" title="c1">a</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c5">br</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <span class="p1"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c4">b</font></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c1">a</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c5">br</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c4">b</font></span>)</span></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="euclid.html#T62" title="EUCLID:th.62">EUCLID:62</a></span>;<br/><span class="kw">then </span><span class="lab"><font color="Green" title="E21">A5</font></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"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><font color="Maroon" title="c1">a</font>,<span class="p4">(<span class="default"><font color="Maroon" title="c5">br</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <span class="p3"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c4">b</font></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> = 
<span class="p1"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c5">br</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c4">b</font></span>)</span></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="topreal6.html#T23" title="TOPREAL6:th.23">TOPREAL6:23</a></span>
<br/>.= 
<span class="p1"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">b</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">br</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="complex1.html#T60" title="COMPLEX1:th.60">COMPLEX1:60</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">b</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c7">r</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2
<span class="kw">by</span> <span class="lab"><a class="ref" href="absvalue.html#D1" title="ABSVALUE:def.1">ABSVALUE:def 1</a></span>
;<br/><a NAME="E9:23_1"/>
<font color="Maroon" title="c4">b</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c7">r</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c7">r</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c7">r</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:23_1"><span class="lab"><font color="Green" title="E20">A4</font></span></a>, <a class="ref" href="xreal_1.html#T6" title="XREAL_1:th.6">XREAL_1:6</a></span>;<br/><a NAME="E10:23_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c4">b</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c7">r</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">r</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c7">r</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T74" title="XREAL_1:th.74">XREAL_1:74</a></span>;<br/><a NAME="E11:23_1"/><span class="kw">then </span>
<span class="p1"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><font color="Maroon" title="c1">a</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c5">br</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal9.html#K1" title="TOPREAL9:func.1">Ball</a> (<span class="p1"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c4">b</font></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span>,<font color="Maroon" title="c7">r</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:23_1"><span class="lab"><font color="Green" title="E21">A5</font></span></a>, <a class="ref" href="topreal9.html#T7" title="TOPREAL9:th.7">TOPREAL9:7</a></span>;<br/><span class="kw">hence </span><a NAME="E12:23_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:23_1"><span class="lab"><font color="Green" title="E19">A3</font></span></a>, <a class="ref" href="topgen_5.html#T18" target="_self" title="TOPGEN_5:th.18">Th22</a></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/></div>
<span class="kw">end;</span></div>

<span class="kw">assume </span><a NAME="E3:23"/><span class="lab"><font color="Green" title="E18">A6</font></span>: 
<font color="Maroon" title="c3">r</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">b</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="topreal9.html#K1" title="TOPREAL9:func.1">Ball</a> (<span class="p1"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span>,<font color="Maroon" title="c3">r</font>) <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="topgen_5.html#K2" title="TOPGEN_5:func.2">y&gt;=0-plane</a> </span><br/>

<span class="kw">let </span><font color="Maroon" title="c4">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal9.html#K1" title="TOPREAL9:func.1">Ball</a> (<span class="p1"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span>,<font color="Maroon" title="c3">r</font>) or <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topgen_5.html#K2" title="TOPGEN_5:func.2">y&gt;=0-plane</a>  )</span><br/>

<span class="kw">assume </span><a NAME="E4:23"/><span class="lab"><font color="Green" title="E19">A7</font></span>: 
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal9.html#K1" title="TOPREAL9:func.1">Ball</a> (<span class="p1"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span>,<font color="Maroon" title="c3">r</font>)
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topgen_5.html#K2" title="TOPGEN_5:func.2">y&gt;=0-plane</a> </span><br/>

<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c5">z</font> = <font color="Maroon" title="c4">x</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> ;<br/>
<a NAME="E6:23"/><span class="lab"><font color="Green" title="E20">A8</font></span>: 
<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">z</font> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <span class="p3"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">r</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:23"><span class="lab"><font color="Green" title="E19">A7</font></span></a>, <a class="ref" href="topreal9.html#T7" title="TOPREAL9:th.7">TOPREAL9:7</a></span>;<br/>
<a NAME="E7:23"/><span class="lab"><font color="Green" title="E21">A9</font></span>: 
<span class="p1"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c5">z</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">a</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c5">z</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">b</font></span>)</span></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">z</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">a</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="euclid.html#T52" title="EUCLID:th.52">EUCLID:52</a></span>;<br/>
<a NAME="E8:23"/><span class="lab"><font color="Green" title="E22">A10</font></span>: 
<span class="p1"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c5">z</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">a</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c5">z</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">b</font></span>)</span></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">z</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">b</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="euclid.html#T52" title="EUCLID:th.52">EUCLID:52</a></span>;<br/>
<a NAME="E9:23"/><span class="lab"><font color="Green" title="E23">A11</font></span>: 
<font color="Maroon" title="c5">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">z</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c5">z</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="euclid.html#T53" title="EUCLID:th.53">EUCLID:53</a></span>;<br/>
<a NAME="E10:23"/><span class="kw">then </span>
<font color="Maroon" title="c5">z</font> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <span class="p1"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c5">z</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">a</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c5">z</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">b</font></span>)</span></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="euclid.html#T62" title="EUCLID:th.62">EUCLID:62</a></span>;<br/>
<a NAME="E11:23"/><span class="kw">then </span>
<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">z</font> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <span class="p3"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="square_1.html#K7" title="SQUARE_1:func.7">sqrt</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c5">z</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">a</font></span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c5">z</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:23"><span class="lab"><font color="Green" title="E21">A9</font></span></a>, <a class="txt" href="#E8:23"><span class="lab"><font color="Green" title="E22">A10</font></span></a>, <a class="ref" href="jgraph_1.html#T30" title="JGRAPH_1:th.30">JGRAPH_1:30</a></span>;<br/>
<a NAME="E12:23"/><span class="kw">then </span>
 <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">z</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">b</font></span>)</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="c5">z</font> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <span class="p3"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="complex1.html#T79" title="COMPLEX1:th.79">COMPLEX1:79</a></span>;<br/>
<a NAME="E13:23"/><span class="kw">then </span>
 <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">z</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">r</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:23"><span class="lab"><font color="Green" title="E20">A8</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/>
<a NAME="E14:23"/><span class="kw">then </span><span class="lab"><font color="Green" title="E24">A12</font></span>: 
 <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">z</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">r</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="complex1.html#T60" title="COMPLEX1:th.60">COMPLEX1:60</a></span>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="23_2"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:23_2"/>
<font color="Maroon" title="c5">z</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><a NAME="E2:23_2"/><span class="kw">then </span>
<font color="Maroon" title="c2">b</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">z</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c2">b</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T46" title="XREAL_1:th.46">XREAL_1:46</a></span>;<br/><a NAME="E3:23_2"/><span class="kw">then </span>
<font color="Maroon" title="c2">b</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">z</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c3">r</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:23"><span class="lab"><font color="Green" title="E18">A6</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><span class="kw">hence </span><a NAME="E4:23_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:23"><span class="lab"><font color="Green" title="E17">A1</font></span></a>, <a class="txt" href="#E14:23"><span class="lab"><font color="Green" title="E24">A12</font></span></a>, <a class="ref" href="absvalue.html#D1" title="ABSVALUE:def.1">ABSVALUE:def 1</a></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/></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E16:23"/>
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topgen_5.html#K2" title="TOPGEN_5:func.2">y&gt;=0-plane</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:23"><span class="lab"><font color="Green" title="E23">A11</font></span></a></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/>


</div>
