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<span class="kw">let </span><font color="Maroon" title="c1">X</font> be   non  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>  <a href="pre_topc.html#NM1" title="PRE_TOPC:NM.1">TopSpace</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">f1</font> being   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c1">X</font>,<a href="topmetr.html#K3" title="TOPMETR:func.3">R^1</a>  st <font color="Olive" title="b1">f1</font> is  <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a>  &amp; (  for <font color="Olive" title="b2">q</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">X</font>  ex <font color="Olive" title="b3">r</font> being   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a> st <br/>( <font color="Olive" title="b1">f1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">r</font> &amp; <font color="Olive" title="b3">r</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  ) ) holds <br/> ex <font color="Olive" title="b2">g</font> being   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c1">X</font>,<a href="topmetr.html#K3" title="TOPMETR:func.3">R^1</a> st <br/>( (  for <font color="Olive" title="b3">p</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">X</font><br/>  for <font color="Olive" title="b4">r1</font> being   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a>  st <font color="Olive" title="b1">f1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b3">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">r1</font> holds <br/><font color="Olive" title="b2">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b3">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="square_1.html#K3" title="SQUARE_1:func.3">sqrt</a> <font color="Olive" title="b4">r1</font> ) &amp; <font color="Olive" title="b2">g</font> is  <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a>  )</span><br/><span class="kw">let </span><font color="Maroon" title="c2">f1</font> be   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c1">X</font>,<a href="topmetr.html#K3" title="TOPMETR:func.3">R^1</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c2">f1</font> is  <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a>  &amp; (  for <font color="Olive" title="b1">q</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">X</font>  ex <font color="Olive" title="b2">r</font> being   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a> st <br/>( <font color="Maroon" title="c2">f1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">r</font> &amp; <font color="Olive" title="b2">r</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  ) ) implies  ex <font color="Olive" title="b1">g</font> being   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c1">X</font>,<a href="topmetr.html#K3" title="TOPMETR:func.3">R^1</a> st <br/>( (  for <font color="Olive" title="b2">p</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">X</font><br/>  for <font color="Olive" title="b3">r1</font> being   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a>  st <font color="Maroon" title="c2">f1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">r1</font> holds <br/><font color="Olive" title="b1">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="square_1.html#K3" title="SQUARE_1:func.3">sqrt</a> <font color="Olive" title="b3">r1</font> ) &amp; <font color="Olive" title="b1">g</font> is  <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a>  ) )</span><br/>



<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:6"/><span class="lab"><font color="Green" title="E8">A1</font></span>: 
<font color="Maroon" title="c2">f1</font> is  <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a> 
 <span class="kw">and </span><br/><a NAME="E2:6"/><span class="lab"><font color="Green" title="E9">A2</font></span>: 
 for <font color="Olive" title="b1">q</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">X</font>  ex <font color="Olive" title="b2">r</font> being   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a> st <br/>( <font color="Maroon" title="c2">f1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">r</font> &amp; <font color="Olive" title="b2">r</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  )
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">g</font> being   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c1">X</font>,<a href="topmetr.html#K3" title="TOPMETR:func.3">R^1</a> st <br/>( (  for <font color="Olive" title="b2">p</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">X</font><br/>  for <font color="Olive" title="b3">r1</font> being   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a>  st <font color="Maroon" title="c2">f1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">r1</font> holds <br/><font color="Olive" title="b1">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="square_1.html#K3" title="SQUARE_1:func.3">sqrt</a> <font color="Olive" title="b3">r1</font> ) &amp; <font color="Olive" title="b1">g</font> is  <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a>  )</span><br/>

<span class="kw">defpred </span><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a> ,   <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a> ] <span class="kw">means </span> for <font color="Olive" title="b1">r11</font> being   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a>  st <font color="Maroon" title="c2">f1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> $1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">r11</font> holds <br/>$2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="square_1.html#K3" title="SQUARE_1:func.3">sqrt</a> <font color="Olive" title="b1">r11</font>;<br/>
<a NAME="E3:6"/><span class="lab"><font color="Green" title="E10">A3</font></span>: 
 for <font color="Olive" title="b1">x</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">X</font>  ex <font color="Olive" title="b2">y</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font>]
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="6_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c3">x</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">X</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">y</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c3">x</font>,<font color="Olive" title="b1">y</font>]</span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c4">r1</font> = <font color="Maroon" title="c2">f1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">x</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  <span class="kw">by</span> <span class="lab"><a class="ref" href="topmetr.html#T17" onmouseover="rs('topmetr/T17')" onmouseout="rh()">TOPMETR:17</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c5">y</font> =  <a href="square_1.html#K3" title="SQUARE_1:func.3">sqrt</a> <font color="Maroon" title="c4">r1</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_0.html#D1" onmouseover="rs('xreal_0/D1')" onmouseout="rh()">XREAL_0:def 1</a></span>;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c5">y</font>
; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c5">y</font>]</span><br/>

<span class="kw">thus </span><a NAME="E3:6_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c5">y</font>]
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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>
<a NAME="E4:6"/>
 ex <font color="Olive" title="b1">f</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">X</font>,<a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> st <br/> for <font color="Olive" title="b2">x2</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">X</font> holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b2">x2</font>,<font color="Olive" title="b1">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">x2</font>]
 
<span class="kw">from</span> <span class="lab"><a class="ref" href="funct_2.html#S3" onmouseover="rs('funct_2/S3')" onmouseout="rh()">FUNCT_2:sch 3</a>(<a class="txt" href="#E3:6"><span class="lab"><font color="Green" title="E10">A3</font></span></a>);<br/></span>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c3">f</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">X</font>,<a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a><span class="kw"> such that </span><br/><a NAME="E6:6"/><span class="lab"><font color="Green" title="E11">A4</font></span>: 
 for <font color="Olive" title="b1">x2</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">X</font><br/>  for <font color="Olive" title="b2">r11</font> being   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a>  st <font color="Maroon" title="c2">f1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">x2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">r11</font> holds <br/><font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">x2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="square_1.html#K3" title="SQUARE_1:func.3">sqrt</a> <font color="Olive" title="b2">r11</font>
 ;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c4">g0</font> = <font color="Maroon" title="c3">f</font> as   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c1">X</font>,<a href="topmetr.html#K3" title="TOPMETR:func.3">R^1</a> <span class="kw">by</span> <span class="lab"><a class="ref" href="topmetr.html#T17" onmouseover="rs('topmetr/T17')" onmouseout="rh()">TOPMETR:17</a></span>;<br/>
<a NAME="E8:6"/>
 for <font color="Olive" title="b1">p</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">X</font><br/>  for <font color="Olive" title="b2">V</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <a href="topmetr.html#K3" title="TOPMETR:func.3">R^1</a>  st <font color="Maroon" title="c4">g0</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">p</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Olive" title="b2">V</font> &amp; <font color="Olive" title="b2">V</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  holds <br/> ex <font color="Olive" title="b3">W</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font> st <br/>( <font color="Olive" title="b1">p</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Olive" title="b3">W</font> &amp; <font color="Olive" title="b3">W</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  &amp; <font color="Maroon" title="c4">g0</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Olive" title="b3">W</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b2">V</font> )
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="6_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c5">p</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">X</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">V</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <a href="topmetr.html#K3" title="TOPMETR:func.3">R^1</a>  st <font color="Maroon" title="c4">g0</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">p</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Olive" title="b1">V</font> &amp; <font color="Olive" title="b1">V</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  holds <br/> ex <font color="Olive" title="b2">W</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font> st <br/>( <font color="Maroon" title="c5">p</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Olive" title="b2">W</font> &amp; <font color="Olive" title="b2">W</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  &amp; <font color="Maroon" title="c4">g0</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Olive" title="b2">W</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">V</font> )</span><br/><span class="kw">let </span><font color="Maroon" title="c6">V</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <a href="topmetr.html#K3" title="TOPMETR:func.3">R^1</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c4">g0</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">p</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c6">V</font> &amp; <font color="Maroon" title="c6">V</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  implies  ex <font color="Olive" title="b1">W</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font> st <br/>( <font color="Maroon" title="c5">p</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Olive" title="b1">W</font> &amp; <font color="Olive" title="b1">W</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  &amp; <font color="Maroon" title="c4">g0</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Olive" title="b1">W</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c6">V</font> ) )</span><br/>



<span class="kw">reconsider </span><font color="Maroon" title="c7">r</font> = <font color="Maroon" title="c4">g0</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">p</font> as   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a> ;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c8">r1</font> = <font color="Maroon" title="c2">f1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">p</font> as   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a> ;<br/>
<span class="kw">assume </span><a NAME="E3:6_2"/>
( <font color="Maroon" title="c4">g0</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">p</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c6">V</font> &amp; <font color="Maroon" title="c6">V</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  )
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">W</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font> st <br/>( <font color="Maroon" title="c5">p</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Olive" title="b1">W</font> &amp; <font color="Olive" title="b1">W</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  &amp; <font color="Maroon" title="c4">g0</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Olive" title="b1">W</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c6">V</font> )</span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c9">r01</font> being   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a><span class="kw"> such that </span><br/><a NAME="E5:6_2"/><span class="lab"><font color="Green" title="E12">A5</font></span>: 
<font color="Maroon" title="c9">r01</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> 
 <span class="kw">and </span><br/><a NAME="E6:6_2"/><span class="lab"><font color="Green" title="E13">A6</font></span>: 
<span class="p1"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">r</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c9">r01</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c7">r</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c9">r01</font></span>)</span></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c6">V</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="frechet.html#T8" onmouseover="rs('frechet/T8')" onmouseout="rh()">FRECHET:8</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c10">r0</font> =  <a href="xxreal_0.html#K3" title="XXREAL_0:func.3">min</a> (<font color="Maroon" title="c9">r01</font>,1);<br/>
<a NAME="E7:6_2"/><span class="lab"><font color="Green" title="E14">A7</font></span>: 
 <a href="xxreal_0.html#K3" title="XXREAL_0:func.3">min</a> (<font color="Maroon" title="c9">r01</font>,1) <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:6_2"><span class="lab"><font color="Green" title="E12">A5</font></span></a>, <a class="ref" href="xxreal_0.html#T21" onmouseover="rs('xxreal_0/T21')" onmouseout="rh()">XXREAL_0:21</a></span>;<br/>
<a NAME="E8:6_2"/><span class="lab"><font color="Green" title="E15">A8</font></span>: 
 <a href="xxreal_0.html#K3" title="XXREAL_0:func.3">min</a> (<font color="Maroon" title="c9">r01</font>,1) <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:6_2"><span class="lab"><font color="Green" title="E12">A5</font></span></a>, <a class="ref" href="xxreal_0.html#T21" onmouseover="rs('xxreal_0/T21')" onmouseout="rh()">XXREAL_0:21</a></span>;<br/>
<a NAME="E9:6_2"/>
 <a href="xxreal_0.html#K3" title="XXREAL_0:func.3">min</a> (<font color="Maroon" title="c9">r01</font>,1) <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c9">r01</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_0.html#T17" onmouseover="rs('xxreal_0/T17')" onmouseout="rh()">XXREAL_0:17</a></span>;<br/>
<a NAME="E10:6_2"/><span class="kw">then </span>
( <font color="Maroon" title="c7">r</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c9">r01</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c7">r</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><a href="xxreal_0.html#K3" title="XXREAL_0:func.3">min</a> (<font color="Maroon" title="c9">r01</font>,1)</span>)</span> &amp; <font color="Maroon" title="c7">r</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><a href="xxreal_0.html#K3" title="XXREAL_0:func.3">min</a> (<font color="Maroon" title="c9">r01</font>,1)</span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c7">r</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c9">r01</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T6" onmouseover="rs('xreal_1/T6')" onmouseout="rh()">XREAL_1:6</a>, <a class="ref" href="xreal_1.html#T10" onmouseover="rs('xreal_1/T10')" onmouseout="rh()">XREAL_1:10</a></span>;<br/>
<a NAME="E11:6_2"/><span class="kw">then </span>
<span class="p1"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">r</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p3">(<span class="default"><a href="xxreal_0.html#K3" title="XXREAL_0:func.3">min</a> (<font color="Maroon" title="c9">r01</font>,1)</span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c7">r</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p3">(<span class="default"><a href="xxreal_0.html#K3" title="XXREAL_0:func.3">min</a> (<font color="Maroon" title="c9">r01</font>,1)</span>)</span></span>)</span></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">r</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c9">r01</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c7">r</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c9">r01</font></span>)</span></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_1.html#T46" onmouseover="rs('xxreal_1/T46')" onmouseout="rh()">XXREAL_1:46</a></span>;<br/>
<a NAME="E12:6_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E16">A9</font></span>: 
<span class="p1"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">r</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p3">(<span class="default"><a href="xxreal_0.html#K3" title="XXREAL_0:func.3">min</a> (<font color="Maroon" title="c9">r01</font>,1)</span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c7">r</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p3">(<span class="default"><a href="xxreal_0.html#K3" title="XXREAL_0:func.3">min</a> (<font color="Maroon" title="c9">r01</font>,1)</span>)</span></span>)</span></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c6">V</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:6_2"><span class="lab"><font color="Green" title="E13">A6</font></span></a></span>;<br/>
<a NAME="E13:6_2"/><span class="lab"><font color="Green" title="E17">A10</font></span>: 
 ex <font color="Olive" title="b1">r8</font> being   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a> st <br/>( <font color="Maroon" title="c2">f1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">r8</font> &amp; <font color="Olive" title="b1">r8</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:6"><span class="lab"><font color="Green" title="E9">A2</font></span></a></span>;<br/>
<a NAME="E14:6_2"/><span class="lab"><font color="Green" title="E18">A11</font></span>: 
<font color="Maroon" title="c7">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="square_1.html#K3" title="SQUARE_1:func.3">sqrt</a> <font color="Maroon" title="c8">r1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:6"><span class="lab"><font color="Green" title="E11">A4</font></span></a></span>;<br/>
<a NAME="E15:6_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E19">A12</font></span>: 
<font color="Maroon" title="c8">r1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">r</font> <a href="square_1.html#K1" title="SQUARE_1:func.1">^2</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:6_2"><span class="lab"><font color="Green" title="E17">A10</font></span></a>, <a class="ref" href="square_1.html#D2" onmouseover="rs('square_1/D2')" onmouseout="rh()">SQUARE_1:def 2</a></span>;<br/>
<a NAME="E16:6_2"/><span class="lab"><font color="Green" title="E20">A13</font></span>: 
<font color="Maroon" title="c7">r</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:6_2"><span class="lab"><font color="Green" title="E17">A10</font></span></a>, <a class="txt" href="#E14:6_2"><span class="lab"><font color="Green" title="E18">A11</font></span></a>, <a class="ref" href="square_1.html#T17" onmouseover="rs('square_1/T17')" onmouseout="rh()">SQUARE_1:17</a>, <a class="ref" href="square_1.html#T26" onmouseover="rs('square_1/T26')" onmouseout="rh()">SQUARE_1:26</a></span>;<br/>
<a NAME="E17:6_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E21">A14</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c7">r</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><a href="xxreal_0.html#K3" title="XXREAL_0:func.3">min</a> (<font color="Maroon" title="c9">r01</font>,1)</span>)</span></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="xxreal_0.html#K3" title="XXREAL_0:func.3">min</a> (<font color="Maroon" title="c9">r01</font>,1)</span>)</span> <a href="square_1.html#K1" title="SQUARE_1:func.1">^2</a></span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:6_2"><span class="lab"><font color="Green" title="E15">A8</font></span></a>, <a class="ref" href="square_1.html#T12" onmouseover="rs('square_1/T12')" onmouseout="rh()">SQUARE_1:12</a>, <a class="ref" href="xreal_1.html#T8" onmouseover="rs('xreal_1/T8')" onmouseout="rh()">XREAL_1:8</a></span>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="6_2_1"><span class="kw">per </span></a><a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:6_2_1"/>
( <font color="Maroon" title="c7">r</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><a href="xxreal_0.html#K3" title="XXREAL_0:func.3">min</a> (<font color="Maroon" title="c9">r01</font>,1)</span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  or <font color="Maroon" title="c7">r</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><a href="xxreal_0.html#K3" title="XXREAL_0:func.3">min</a> (<font color="Maroon" title="c9">r01</font>,1)</span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="6_2_1_1"><span class="kw">suppose </span></a><a NAME="E1:6_2_1_1"/><span class="lab"><font color="Green" title="E22">A15</font></span>: 
<font color="Maroon" title="c7">r</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><a href="xxreal_0.html#K3" title="XXREAL_0:func.3">min</a> (<font color="Maroon" title="c9">r01</font>,1)</span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> 
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">W</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font> st <br/>( <font color="Maroon" title="c5">p</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Olive" title="b1">W</font> &amp; <font color="Olive" title="b1">W</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  &amp; <font color="Maroon" title="c4">g0</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Olive" title="b1">W</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c6">V</font> )</span><br/><div class="add"><span class="kw">set </span><font color="Maroon" title="c11">r4</font> = <span class="p1">(<span class="default"><a href="xxreal_0.html#K3" title="XXREAL_0:func.3">min</a> (<font color="Maroon" title="c9">r01</font>,1)</span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">r</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p2">(<span class="default"><a href="xxreal_0.html#K3" title="XXREAL_0:func.3">min</a> (<font color="Maroon" title="c9">r01</font>,1)</span>)</span></span>)</span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c12">G1</font> = <span class="p1"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">r1</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="xxreal_0.html#K3" title="XXREAL_0:func.3">min</a> (<font color="Maroon" title="c9">r01</font>,1)</span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c7">r</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p5">(<span class="default"><a href="xxreal_0.html#K3" title="XXREAL_0:func.3">min</a> (<font color="Maroon" title="c9">r01</font>,1)</span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c8">r1</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="xxreal_0.html#K3" title="XXREAL_0:func.3">min</a> (<font color="Maroon" title="c9">r01</font>,1)</span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c7">r</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p5">(<span class="default"><a href="xxreal_0.html#K3" title="XXREAL_0:func.3">min</a> (<font color="Maroon" title="c9">r01</font>,1)</span>)</span></span>)</span></span>)</span></span>)</span></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <a href="topmetr.html#K3" title="TOPMETR:func.3">R^1</a> <span class="kw">by</span> <span class="lab"><a class="ref" href="topmetr.html#T17" onmouseover="rs('topmetr/T17')" onmouseout="rh()">TOPMETR:17</a></span>;<br/><a NAME="E3:6_2_1_1"/><span class="lab"><font color="Green" title="E23">A16</font></span>: 
<font color="Maroon" title="c8">r1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c8">r1</font> <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="xxreal_0.html#K3" title="XXREAL_0:func.3">min</a> (<font color="Maroon" title="c9">r01</font>,1)</span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">r</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p3">(<span class="default"><a href="xxreal_0.html#K3" title="XXREAL_0:func.3">min</a> (<font color="Maroon" title="c9">r01</font>,1)</span>)</span></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:6_2"><span class="lab"><font color="Green" title="E15">A8</font></span></a>, <a class="txt" href="#E1:6_2_1_1"><span class="lab"><font color="Green" title="E22">A15</font></span></a>, <a class="ref" href="xreal_1.html#T29" onmouseover="rs('xreal_1/T29')" onmouseout="rh()">XREAL_1:29</a>, <a class="ref" href="xreal_1.html#T129" onmouseover="rs('xreal_1/T129')" onmouseout="rh()">XREAL_1:129</a></span>;<br/><a NAME="E4:6_2_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c8">r1</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="xxreal_0.html#K3" title="XXREAL_0:func.3">min</a> (<font color="Maroon" title="c9">r01</font>,1)</span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">r</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p3">(<span class="default"><a href="xxreal_0.html#K3" title="XXREAL_0:func.3">min</a> (<font color="Maroon" title="c9">r01</font>,1)</span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c8">r1</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T19" onmouseover="rs('xreal_1/T19')" onmouseout="rh()">XREAL_1:19</a></span>;<br/><a NAME="E5:6_2_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E24">A17</font></span>: 
<font color="Maroon" title="c2">f1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">p</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c12">G1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:6_2_1_1"><span class="lab"><font color="Green" title="E23">A16</font></span></a>, <a class="ref" href="xxreal_1.html#T4" onmouseover="rs('xxreal_1/T4')" onmouseout="rh()">XXREAL_1:4</a></span>;<br/><a NAME="E6:6_2_1_1"/>
<font color="Maroon" title="c12">G1</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="jordan6.html#T35" onmouseover="rs('jordan6/T35')" onmouseout="rh()">JORDAN6:35</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c13">W1</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font><span class="kw"> such that </span><br/><a NAME="E8:6_2_1_1"/><span class="lab"><font color="Green" title="E25">A18</font></span>: 
( <font color="Maroon" title="c5">p</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c13">W1</font> &amp; <font color="Maroon" title="c13">W1</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  )
 <span class="kw">and </span><br/><a NAME="E9:6_2_1_1"/><span class="lab"><font color="Green" title="E26">A19</font></span>: 
<font color="Maroon" title="c2">f1</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c13">W1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c12">G1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:6"><span class="lab"><font color="Green" title="E8">A1</font></span></a>, <a class="txt" href="#E5:6_2_1_1"><span class="lab"><font color="Green" title="E24">A17</font></span></a>, <a class="ref" href="jgraph_2.html#T10" onmouseover="rs('jgraph_2/T10')" onmouseout="rh()">JGRAPH_2:10</a></span>;<br/><span class="kw">set </span><font color="Maroon" title="c14">W</font> = <font color="Maroon" title="c13">W1</font>;<br/><a NAME="E10:6_2_1_1"/><span class="lab"><font color="Green" title="E27">A20</font></span>: 
( <span class="p1">(<span class="default"><font color="Maroon" title="c7">r</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2</span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p3">(<span class="default"><a href="xxreal_0.html#K3" title="XXREAL_0:func.3">min</a> (<font color="Maroon" title="c9">r01</font>,1)</span>)</span></span>)</span></span>)</span> <a href="square_1.html#K1" title="SQUARE_1:func.1">^2</a>  <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  &amp; <span class="p1">(<span class="default"><a href="xxreal_0.html#K3" title="XXREAL_0:func.3">min</a> (<font color="Maroon" title="c9">r01</font>,1)</span>)</span> <a href="square_1.html#K1" title="SQUARE_1:func.1">^2</a>  <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T63" onmouseover="rs('xreal_1/T63')" onmouseout="rh()">XREAL_1:63</a></span>;<br/><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="6_2_1_1_1"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  not <font color="Maroon" title="c8">r1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:6_2_1_1_1"/>
<font color="Maroon" title="c8">r1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> 
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><a NAME="E2:6_2_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c7">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:6"><span class="lab"><font color="Green" title="E11">A4</font></span></a>, <a class="ref" href="square_1.html#T17" onmouseover="rs('square_1/T17')" onmouseout="rh()">SQUARE_1:17</a></span>;<br/><span class="kw">hence </span><a NAME="E3:6_2_1_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:6_2"><span class="lab"><font color="Green" title="E14">A7</font></span></a>, <a class="txt" href="#E1:6_2_1_1"><span class="lab"><font color="Green" title="E22">A15</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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><a NAME="E12:6_2_1_1"/><span class="kw">then </span>
 <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c7">r</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:6_2"><span class="lab"><font color="Green" title="E17">A10</font></span></a>, <a class="txt" href="#E14:6_2"><span class="lab"><font color="Green" title="E18">A11</font></span></a>, <a class="ref" href="square_1.html#T25" onmouseover="rs('square_1/T25')" onmouseout="rh()">SQUARE_1:25</a></span>;<br/><a NAME="E13:6_2_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E28">A21</font></span>: 
<span class="p1">(<span class="default"><a href="xxreal_0.html#K3" title="XXREAL_0:func.3">min</a> (<font color="Maroon" title="c9">r01</font>,1)</span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c7">r</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:6_2"><span class="lab"><font color="Green" title="E15">A8</font></span></a>, <a class="ref" href="xreal_1.html#T129" onmouseover="rs('xreal_1/T129')" onmouseout="rh()">XREAL_1:129</a></span>;<br/><a NAME="E14:6_2_1_1"/><span class="kw">then </span>
<a href="numbers.html#K5" title="NUMBERS:func.5">0</a> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">r</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><a href="xxreal_0.html#K3" title="XXREAL_0:func.3">min</a> (<font color="Maroon" title="c9">r01</font>,1)</span>)</span></span>)</span> <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="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><a href="xxreal_0.html#K3" title="XXREAL_0:func.3">min</a> (<font color="Maroon" title="c9">r01</font>,1)</span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">r</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><a href="xxreal_0.html#K3" title="XXREAL_0:func.3">min</a> (<font color="Maroon" title="c9">r01</font>,1)</span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T8" onmouseover="rs('xreal_1/T8')" onmouseout="rh()">XREAL_1:8</a></span>;<br/><a NAME="E15:6_2_1_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="xxreal_0.html#K3" title="XXREAL_0:func.3">min</a> (<font color="Maroon" title="c9">r01</font>,1)</span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c7">r</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="xxreal_0.html#K3" title="XXREAL_0:func.3">min</a> (<font color="Maroon" title="c9">r01</font>,1)</span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><a href="xxreal_0.html#K3" title="XXREAL_0:func.3">min</a> (<font color="Maroon" title="c9">r01</font>,1)</span>)</span></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c7">r</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><a href="xxreal_0.html#K3" title="XXREAL_0:func.3">min</a> (<font color="Maroon" title="c9">r01</font>,1)</span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="xxreal_0.html#K3" title="XXREAL_0:func.3">min</a> (<font color="Maroon" title="c9">r01</font>,1)</span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><a href="xxreal_0.html#K3" title="XXREAL_0:func.3">min</a> (<font color="Maroon" title="c9">r01</font>,1)</span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T14" onmouseover="rs('xreal_1/T14')" onmouseout="rh()">XREAL_1:14</a></span>;<br/><a NAME="E16:6_2_1_1"/><span class="kw">then </span>
 <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="xxreal_0.html#K3" title="XXREAL_0:func.3">min</a> (<font color="Maroon" title="c9">r01</font>,1)</span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">r</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p3">(<span class="default"><a href="xxreal_0.html#K3" title="XXREAL_0:func.3">min</a> (<font color="Maroon" title="c9">r01</font>,1)</span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c7">r</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p3">(<span class="default"><a href="xxreal_0.html#K3" title="XXREAL_0:func.3">min</a> (<font color="Maroon" title="c9">r01</font>,1)</span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="xxreal_0.html#K3" title="XXREAL_0:func.3">min</a> (<font color="Maroon" title="c9">r01</font>,1)</span>)</span> <a href="square_1.html#K1" title="SQUARE_1:func.1">^2</a></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T24" onmouseover="rs('xreal_1/T24')" onmouseout="rh()">XREAL_1:24</a></span>;<br/><a NAME="E17:6_2_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c8">r1</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="xxreal_0.html#K3" title="XXREAL_0:func.3">min</a> (<font color="Maroon" title="c9">r01</font>,1)</span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">r</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p4">(<span class="default"><a href="xxreal_0.html#K3" title="XXREAL_0:func.3">min</a> (<font color="Maroon" title="c9">r01</font>,1)</span>)</span></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">r</font> <a href="square_1.html#K1" title="SQUARE_1:func.1">^2</a></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c7">r</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p4">(<span class="default"><a href="xxreal_0.html#K3" title="XXREAL_0:func.3">min</a> (<font color="Maroon" title="c9">r01</font>,1)</span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="xxreal_0.html#K3" title="XXREAL_0:func.3">min</a> (<font color="Maroon" title="c9">r01</font>,1)</span>)</span> <a href="square_1.html#K1" title="SQUARE_1:func.1">^2</a></span>)</span></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:6_2"><span class="lab"><font color="Green" title="E19">A12</font></span></a>, <a class="ref" href="xreal_1.html#T8" onmouseover="rs('xreal_1/T8')" onmouseout="rh()">XREAL_1:8</a></span>;<br/><a NAME="E18:6_2_1_1"/><span class="kw">then </span>
 <a href="square_1.html#K3" title="SQUARE_1:func.3">sqrt</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">r1</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="xxreal_0.html#K3" title="XXREAL_0:func.3">min</a> (<font color="Maroon" title="c9">r01</font>,1)</span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">r</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p4">(<span class="default"><a href="xxreal_0.html#K3" title="XXREAL_0:func.3">min</a> (<font color="Maroon" title="c9">r01</font>,1)</span>)</span></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="square_1.html#K3" title="SQUARE_1:func.3">sqrt</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">r</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p3">(<span class="default"><a href="xxreal_0.html#K3" title="XXREAL_0:func.3">min</a> (<font color="Maroon" title="c9">r01</font>,1)</span>)</span></span>)</span> <a href="square_1.html#K1" title="SQUARE_1:func.1">^2</a></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="square_1.html#T27" onmouseover="rs('square_1/T27')" onmouseout="rh()">SQUARE_1:27</a>, <a class="ref" href="xreal_1.html#T63" onmouseover="rs('xreal_1/T63')" onmouseout="rh()">XREAL_1:63</a></span>;<br/><a NAME="E19:6_2_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E29">A22</font></span>: 
 <a href="square_1.html#K3" title="SQUARE_1:func.3">sqrt</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">r1</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="xxreal_0.html#K3" title="XXREAL_0:func.3">min</a> (<font color="Maroon" title="c9">r01</font>,1)</span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">r</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p4">(<span class="default"><a href="xxreal_0.html#K3" title="XXREAL_0:func.3">min</a> (<font color="Maroon" title="c9">r01</font>,1)</span>)</span></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c7">r</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><a href="xxreal_0.html#K3" title="XXREAL_0:func.3">min</a> (<font color="Maroon" title="c9">r01</font>,1)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:6_2_1_1"><span class="lab"><font color="Green" title="E22">A15</font></span></a>, <a class="ref" href="square_1.html#T22" onmouseover="rs('square_1/T22')" onmouseout="rh()">SQUARE_1:22</a></span>;<br/><a NAME="E20:6_2_1_1"/>
<a href="numbers.html#K5" title="NUMBERS:func.5">0</a> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">r</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><a href="xxreal_0.html#K3" title="XXREAL_0:func.3">min</a> (<font color="Maroon" title="c9">r01</font>,1)</span>)</span></span>)</span> <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="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><a href="xxreal_0.html#K3" title="XXREAL_0:func.3">min</a> (<font color="Maroon" title="c9">r01</font>,1)</span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">r</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><a href="xxreal_0.html#K3" title="XXREAL_0:func.3">min</a> (<font color="Maroon" title="c9">r01</font>,1)</span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:6_2_1_1"><span class="lab"><font color="Green" title="E28">A21</font></span></a>, <a class="ref" href="xreal_1.html#T8" onmouseover="rs('xreal_1/T8')" onmouseout="rh()">XREAL_1:8</a></span>;<br/><a NAME="E21:6_2_1_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="xxreal_0.html#K3" title="XXREAL_0:func.3">min</a> (<font color="Maroon" title="c9">r01</font>,1)</span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c7">r</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c7">r</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><a href="xxreal_0.html#K3" title="XXREAL_0:func.3">min</a> (<font color="Maroon" title="c9">r01</font>,1)</span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default">2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="xxreal_0.html#K3" title="XXREAL_0:func.3">min</a> (<font color="Maroon" title="c9">r01</font>,1)</span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p3">(<span class="default"><a href="xxreal_0.html#K3" title="XXREAL_0:func.3">min</a> (<font color="Maroon" title="c9">r01</font>,1)</span>)</span></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:6_2"><span class="lab"><font color="Green" title="E15">A8</font></span></a>, <a class="ref" href="xreal_1.html#T8" onmouseover="rs('xreal_1/T8')" onmouseout="rh()">XREAL_1:8</a></span>;<br/><a NAME="E22:6_2_1_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="xxreal_0.html#K3" title="XXREAL_0:func.3">min</a> (<font color="Maroon" title="c9">r01</font>,1)</span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c7">r</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="xxreal_0.html#K3" title="XXREAL_0:func.3">min</a> (<font color="Maroon" title="c9">r01</font>,1)</span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p3">(<span class="default"><a href="xxreal_0.html#K3" title="XXREAL_0:func.3">min</a> (<font color="Maroon" title="c9">r01</font>,1)</span>)</span></span>)</span></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="xxreal_0.html#K3" title="XXREAL_0:func.3">min</a> (<font color="Maroon" title="c9">r01</font>,1)</span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><a href="xxreal_0.html#K3" title="XXREAL_0:func.3">min</a> (<font color="Maroon" title="c9">r01</font>,1)</span>)</span></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c7">r</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p3">(<span class="default"><a href="xxreal_0.html#K3" title="XXREAL_0:func.3">min</a> (<font color="Maroon" title="c9">r01</font>,1)</span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="xxreal_0.html#K3" title="XXREAL_0:func.3">min</a> (<font color="Maroon" title="c9">r01</font>,1)</span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p3">(<span class="default"><a href="xxreal_0.html#K3" title="XXREAL_0:func.3">min</a> (<font color="Maroon" title="c9">r01</font>,1)</span>)</span></span>)</span></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="xxreal_0.html#K3" title="XXREAL_0:func.3">min</a> (<font color="Maroon" title="c9">r01</font>,1)</span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><a href="xxreal_0.html#K3" title="XXREAL_0:func.3">min</a> (<font color="Maroon" title="c9">r01</font>,1)</span>)</span></span>)</span>
 ;<br/><a NAME="E23:6_2_1_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="xxreal_0.html#K3" title="XXREAL_0:func.3">min</a> (<font color="Maroon" title="c9">r01</font>,1)</span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c7">r</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="xxreal_0.html#K3" title="XXREAL_0:func.3">min</a> (<font color="Maroon" title="c9">r01</font>,1)</span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><a href="xxreal_0.html#K3" title="XXREAL_0:func.3">min</a> (<font color="Maroon" title="c9">r01</font>,1)</span>)</span></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c7">r</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><a href="xxreal_0.html#K3" title="XXREAL_0:func.3">min</a> (<font color="Maroon" title="c9">r01</font>,1)</span>)</span></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="xxreal_0.html#K3" title="XXREAL_0:func.3">min</a> (<font color="Maroon" title="c9">r01</font>,1)</span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><a href="xxreal_0.html#K3" title="XXREAL_0:func.3">min</a> (<font color="Maroon" title="c9">r01</font>,1)</span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T7" onmouseover="rs('xreal_1/T7')" onmouseout="rh()">XREAL_1:7</a></span>;<br/><a NAME="E24:6_2_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c8">r1</font> <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="xxreal_0.html#K3" title="XXREAL_0:func.3">min</a> (<font color="Maroon" title="c9">r01</font>,1)</span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">r</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p3">(<span class="default"><a href="xxreal_0.html#K3" title="XXREAL_0:func.3">min</a> (<font color="Maroon" title="c9">r01</font>,1)</span>)</span></span>)</span></span>)</span> <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="square_1.html#K1" title="SQUARE_1:func.1">^2</a></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"><span class="p3">(<span class="default">2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c7">r</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p3">(<span class="default"><a href="xxreal_0.html#K3" title="XXREAL_0:func.3">min</a> (<font color="Maroon" title="c9">r01</font>,1)</span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="xxreal_0.html#K3" title="XXREAL_0:func.3">min</a> (<font color="Maroon" title="c9">r01</font>,1)</span>)</span> <a href="square_1.html#K1" title="SQUARE_1:func.1">^2</a></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:6_2"><span class="lab"><font color="Green" title="E19">A12</font></span></a>, <a class="ref" href="xreal_1.html#T8" onmouseover="rs('xreal_1/T8')" onmouseout="rh()">XREAL_1:8</a></span>;<br/><a NAME="E25:6_2_1_1"/><span class="kw">then </span>
 <a href="square_1.html#K3" title="SQUARE_1:func.3">sqrt</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">r1</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="xxreal_0.html#K3" title="XXREAL_0:func.3">min</a> (<font color="Maroon" title="c9">r01</font>,1)</span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">r</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p4">(<span class="default"><a href="xxreal_0.html#K3" title="XXREAL_0:func.3">min</a> (<font color="Maroon" title="c9">r01</font>,1)</span>)</span></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="square_1.html#K3" title="SQUARE_1:func.3">sqrt</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">r</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p3">(<span class="default"><a href="xxreal_0.html#K3" title="XXREAL_0:func.3">min</a> (<font color="Maroon" title="c9">r01</font>,1)</span>)</span></span>)</span> <a href="square_1.html#K1" title="SQUARE_1:func.1">^2</a></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:6_2"><span class="lab"><font color="Green" title="E17">A10</font></span></a>, <a class="txt" href="#E8:6_2"><span class="lab"><font color="Green" title="E15">A8</font></span></a>, <a class="txt" href="#E1:6_2_1_1"><span class="lab"><font color="Green" title="E22">A15</font></span></a>, <a class="ref" href="square_1.html#T27" onmouseover="rs('square_1/T27')" onmouseout="rh()">SQUARE_1:27</a></span>;<br/><a NAME="E26:6_2_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E30">A23</font></span>: 
<font color="Maroon" title="c7">r</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><a href="xxreal_0.html#K3" title="XXREAL_0:func.3">min</a> (<font color="Maroon" title="c9">r01</font>,1)</span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="square_1.html#K3" title="SQUARE_1:func.3">sqrt</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">r1</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="xxreal_0.html#K3" title="XXREAL_0:func.3">min</a> (<font color="Maroon" title="c9">r01</font>,1)</span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">r</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p4">(<span class="default"><a href="xxreal_0.html#K3" title="XXREAL_0:func.3">min</a> (<font color="Maroon" title="c9">r01</font>,1)</span>)</span></span>)</span></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:6_2"><span class="lab"><font color="Green" title="E20">A13</font></span></a>, <a class="txt" href="#E7:6_2"><span class="lab"><font color="Green" title="E14">A7</font></span></a>, <a class="ref" href="square_1.html#T22" onmouseover="rs('square_1/T22')" onmouseout="rh()">SQUARE_1:22</a></span>;<br/><span class="lab"><font color="Green" title="E31">A24</font></span>: <font color="Maroon" title="c8">r1</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="xxreal_0.html#K3" title="XXREAL_0:func.3">min</a> (<font color="Maroon" title="c9">r01</font>,1)</span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">r</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p3">(<span class="default"><a href="xxreal_0.html#K3" title="XXREAL_0:func.3">min</a> (<font color="Maroon" title="c9">r01</font>,1)</span>)</span></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c7">r</font> <a href="square_1.html#K1" title="SQUARE_1:func.1">^2</a></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="xxreal_0.html#K3" title="XXREAL_0:func.3">min</a> (<font color="Maroon" title="c9">r01</font>,1)</span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c7">r</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="xxreal_0.html#K3" title="XXREAL_0:func.3">min</a> (<font color="Maroon" title="c9">r01</font>,1)</span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p3">(<span class="default"><a href="xxreal_0.html#K3" title="XXREAL_0:func.3">min</a> (<font color="Maroon" title="c9">r01</font>,1)</span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:6_2"><span class="lab"><font color="Green" title="E17">A10</font></span></a>, <a class="txt" href="#E14:6_2"><span class="lab"><font color="Green" title="E18">A11</font></span></a>, <a class="ref" href="square_1.html#D2" onmouseover="rs('square_1/D2')" onmouseout="rh()">SQUARE_1:def 2</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">r</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">1 <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2</span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p4">(<span class="default"><a href="xxreal_0.html#K3" title="XXREAL_0:func.3">min</a> (<font color="Maroon" title="c9">r01</font>,1)</span>)</span></span>)</span></span>)</span> <a href="square_1.html#K1" title="SQUARE_1:func.1">^2</a></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">3 <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 4</span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="xxreal_0.html#K3" title="XXREAL_0:func.3">min</a> (<font color="Maroon" title="c9">r01</font>,1)</span>)</span> <a href="square_1.html#K1" title="SQUARE_1:func.1">^2</a></span>)</span></span>)</span>

;<br/><a NAME="E28:6_2_1_1"/>
<font color="Maroon" title="c4">g0</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c13">W1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">r</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p3">(<span class="default"><a href="xxreal_0.html#K3" title="XXREAL_0:func.3">min</a> (<font color="Maroon" title="c9">r01</font>,1)</span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c7">r</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p3">(<span class="default"><a href="xxreal_0.html#K3" title="XXREAL_0:func.3">min</a> (<font color="Maroon" title="c9">r01</font>,1)</span>)</span></span>)</span></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span>
 <div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="6_2_1_1_3"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c15">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="tarski.html#D3" onmouseover="rs('tarski/D3')" onmouseout="rh()">TARSKI:def 3</a> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c15">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">g0</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c13">W1</font> or <font color="Maroon" title="c15">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">r</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p3">(<span class="default"><a href="xxreal_0.html#K3" title="XXREAL_0:func.3">min</a> (<font color="Maroon" title="c9">r01</font>,1)</span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c7">r</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p3">(<span class="default"><a href="xxreal_0.html#K3" title="XXREAL_0:func.3">min</a> (<font color="Maroon" title="c9">r01</font>,1)</span>)</span></span>)</span></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span> )</span><br/>

<span class="kw">assume </span><a NAME="E1:6_2_1_1_3"/>
<font color="Maroon" title="c15">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">g0</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c13">W1</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c15">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">r</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p3">(<span class="default"><a href="xxreal_0.html#K3" title="XXREAL_0:func.3">min</a> (<font color="Maroon" title="c9">r01</font>,1)</span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c7">r</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p3">(<span class="default"><a href="xxreal_0.html#K3" title="XXREAL_0:func.3">min</a> (<font color="Maroon" title="c9">r01</font>,1)</span>)</span></span>)</span></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span></span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c16">z</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> <span class="kw"> such that </span><br/><a NAME="E3:6_2_1_1_3"/><span class="lab"><font color="Green" title="E32">A25</font></span>: 
<font color="Maroon" title="c16">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c4">g0</font>
 <span class="kw">and </span><br/><a NAME="E4:6_2_1_1_3"/><span class="lab"><font color="Green" title="E33">A26</font></span>: 
<font color="Maroon" title="c16">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c13">W1</font>
 <span class="kw">and </span><br/><a NAME="E5:6_2_1_1_3"/><span class="lab"><font color="Green" title="E34">A27</font></span>: 
<font color="Maroon" title="c4">g0</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">x</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#D6" onmouseover="rs('funct_1/D6')" onmouseout="rh()">FUNCT_1:def 6</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c17">pz</font> = <font color="Maroon" title="c16">z</font> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">X</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:6_2_1_1_3"><span class="lab"><font color="Green" title="E32">A25</font></span></a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c18">aa1</font> = <font color="Maroon" title="c2">f1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c17">pz</font> as   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a> ;<br/>
<a NAME="E8:6_2_1_1_3"/><span class="lab"><font color="Green" title="E35">A28</font></span>: 
 ex <font color="Olive" title="b1">r9</font> being   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a> st <br/>( <font color="Maroon" title="c2">f1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c17">pz</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">r9</font> &amp; <font color="Olive" title="b1">r9</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:6"><span class="lab"><font color="Green" title="E9">A2</font></span></a></span>;<br/>
<a NAME="E9:6_2_1_1_3"/>
<font color="Maroon" title="c17">pz</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">X</font>
 
;<br/>
<a NAME="E10:6_2_1_1_3"/><span class="kw">then </span>
<font color="Maroon" title="c17">pz</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c2">f1</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D1" onmouseover="rs('funct_2/D1')" onmouseout="rh()">FUNCT_2:def 1</a></span>;<br/>
<a NAME="E11:6_2_1_1_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E36">A29</font></span>: 
<font color="Maroon" title="c2">f1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c17">pz</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c2">f1</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c13">W1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:6_2_1_1_3"><span class="lab"><font color="Green" title="E33">A26</font></span></a>, <a class="ref" href="funct_1.html#D6" onmouseover="rs('funct_1/D6')" onmouseout="rh()">FUNCT_1:def 6</a></span>;<br/>
<a NAME="E12:6_2_1_1_3"/><span class="kw">then </span>
<font color="Maroon" title="c18">aa1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c8">r1</font> <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="xxreal_0.html#K3" title="XXREAL_0:func.3">min</a> (<font color="Maroon" title="c9">r01</font>,1)</span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">r</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p3">(<span class="default"><a href="xxreal_0.html#K3" title="XXREAL_0:func.3">min</a> (<font color="Maroon" title="c9">r01</font>,1)</span>)</span></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:6_2_1_1"><span class="lab"><font color="Green" title="E26">A19</font></span></a>, <a class="ref" href="xxreal_1.html#T4" onmouseover="rs('xxreal_1/T4')" onmouseout="rh()">XXREAL_1:4</a></span>;<br/>
<a NAME="E13:6_2_1_1_3"/><span class="kw">then </span>
 <a href="square_1.html#K3" title="SQUARE_1:func.3">sqrt</a> <font color="Maroon" title="c18">aa1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="square_1.html#K3" title="SQUARE_1:func.3">sqrt</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">r1</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="xxreal_0.html#K3" title="XXREAL_0:func.3">min</a> (<font color="Maroon" title="c9">r01</font>,1)</span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">r</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p4">(<span class="default"><a href="xxreal_0.html#K3" title="XXREAL_0:func.3">min</a> (<font color="Maroon" title="c9">r01</font>,1)</span>)</span></span>)</span></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:6_2_1_1_3"><span class="lab"><font color="Green" title="E35">A28</font></span></a>, <a class="ref" href="square_1.html#T27" onmouseover="rs('square_1/T27')" onmouseout="rh()">SQUARE_1:27</a></span>;<br/>
<a NAME="E14:6_2_1_1_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E37">A30</font></span>: 
 <a href="square_1.html#K3" title="SQUARE_1:func.3">sqrt</a> <font color="Maroon" title="c18">aa1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c7">r</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><a href="xxreal_0.html#K3" title="XXREAL_0:func.3">min</a> (<font color="Maroon" title="c9">r01</font>,1)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E26:6_2_1_1"><span class="lab"><font color="Green" title="E30">A23</font></span></a>, <a class="ref" href="xxreal_0.html#T2" onmouseover="rs('xxreal_0/T2')" onmouseout="rh()">XXREAL_0:2</a></span>;<br/>
<a NAME="E15:6_2_1_1_3"/><span class="lab"><font color="Green" title="E38">A31</font></span>: 
<font color="Maroon" title="c8">r1</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="xxreal_0.html#K3" title="XXREAL_0:func.3">min</a> (<font color="Maroon" title="c9">r01</font>,1)</span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">r</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p3">(<span class="default"><a href="xxreal_0.html#K3" title="XXREAL_0:func.3">min</a> (<font color="Maroon" title="c9">r01</font>,1)</span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c18">aa1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:6_2_1_1"><span class="lab"><font color="Green" title="E26">A19</font></span></a>, <a class="txt" href="#E11:6_2_1_1_3"><span class="lab"><font color="Green" title="E36">A29</font></span></a>, <a class="ref" href="xxreal_1.html#T4" onmouseover="rs('xxreal_1/T4')" onmouseout="rh()">XXREAL_1:4</a></span>;<br/>
<div><span class="lab"><font color="Green" title="E39">A32</font></span>: <a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="6_2_1_1_3_1"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( (  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c8">r1</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="xxreal_0.html#K3" title="XXREAL_0:func.3">min</a> (<font color="Maroon" title="c9">r01</font>,1)</span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">r</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p3">(<span class="default"><a href="xxreal_0.html#K3" title="XXREAL_0:func.3">min</a> (<font color="Maroon" title="c9">r01</font>,1)</span>)</span></span>)</span></span>)</span> &amp; <font color="Maroon" title="c7">r</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><a href="xxreal_0.html#K3" title="XXREAL_0:func.3">min</a> (<font color="Maroon" title="c9">r01</font>,1)</span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="square_1.html#K3" title="SQUARE_1:func.3">sqrt</a> <font color="Maroon" title="c18">aa1</font> ) or (  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c8">r1</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="xxreal_0.html#K3" title="XXREAL_0:func.3">min</a> (<font color="Maroon" title="c9">r01</font>,1)</span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">r</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p3">(<span class="default"><a href="xxreal_0.html#K3" title="XXREAL_0:func.3">min</a> (<font color="Maroon" title="c9">r01</font>,1)</span>)</span></span>)</span></span>)</span> &amp; contradiction ) )</span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="6_2_1_1_3_1_1"><span class="kw">per </span></a><a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:6_2_1_1_3_1_1"/>
(  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c8">r1</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="xxreal_0.html#K3" title="XXREAL_0:func.3">min</a> (<font color="Maroon" title="c9">r01</font>,1)</span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">r</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p3">(<span class="default"><a href="xxreal_0.html#K3" title="XXREAL_0:func.3">min</a> (<font color="Maroon" title="c9">r01</font>,1)</span>)</span></span>)</span></span>)</span> or  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c8">r1</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="xxreal_0.html#K3" title="XXREAL_0:func.3">min</a> (<font color="Maroon" title="c9">r01</font>,1)</span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">r</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p3">(<span class="default"><a href="xxreal_0.html#K3" title="XXREAL_0:func.3">min</a> (<font color="Maroon" title="c9">r01</font>,1)</span>)</span></span>)</span></span>)</span> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="6_2_1_1_3_1_1_1"><span class="kw">case </span></a><a NAME="E1:6_2_1_1_3_1_1_1"/>
 <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c8">r1</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="xxreal_0.html#K3" title="XXREAL_0:func.3">min</a> (<font color="Maroon" title="c9">r01</font>,1)</span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">r</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p3">(<span class="default"><a href="xxreal_0.html#K3" title="XXREAL_0:func.3">min</a> (<font color="Maroon" title="c9">r01</font>,1)</span>)</span></span>)</span></span>)</span>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c7">r</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><a href="xxreal_0.html#K3" title="XXREAL_0:func.3">min</a> (<font color="Maroon" title="c9">r01</font>,1)</span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="square_1.html#K3" title="SQUARE_1:func.3">sqrt</a> <font color="Maroon" title="c18">aa1</font></span><br/><div class="add"><a NAME="E2:6_2_1_1_3_1_1_1"/><span class="kw">then </span>
 <a href="square_1.html#K3" title="SQUARE_1:func.3">sqrt</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">r1</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="xxreal_0.html#K3" title="XXREAL_0:func.3">min</a> (<font color="Maroon" title="c9">r01</font>,1)</span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">r</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p4">(<span class="default"><a href="xxreal_0.html#K3" title="XXREAL_0:func.3">min</a> (<font color="Maroon" title="c9">r01</font>,1)</span>)</span></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="square_1.html#K3" title="SQUARE_1:func.3">sqrt</a> <font color="Maroon" title="c18">aa1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:6_2_1_1_3"><span class="lab"><font color="Green" title="E38">A31</font></span></a>, <a class="ref" href="square_1.html#T26" onmouseover="rs('square_1/T26')" onmouseout="rh()">SQUARE_1:26</a></span>;<br/><span class="kw">hence </span><a NAME="E3:6_2_1_1_3_1_1_1"/>
<font color="Maroon" title="c7">r</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><a href="xxreal_0.html#K3" title="XXREAL_0:func.3">min</a> (<font color="Maroon" title="c9">r01</font>,1)</span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="square_1.html#K3" title="SQUARE_1:func.3">sqrt</a> <font color="Maroon" title="c18">aa1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:6_2_1_1"><span class="lab"><font color="Green" title="E29">A22</font></span></a>, <a class="ref" href="xxreal_0.html#T2" onmouseover="rs('xxreal_0/T2')" onmouseout="rh()">XXREAL_0:2</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="6_2_1_1_3_1_1_2"><span class="kw">case </span></a><a NAME="E1:6_2_1_1_3_1_1_2"/>
 <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c8">r1</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="xxreal_0.html#K3" title="XXREAL_0:func.3">min</a> (<font color="Maroon" title="c9">r01</font>,1)</span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">r</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p3">(<span class="default"><a href="xxreal_0.html#K3" title="XXREAL_0:func.3">min</a> (<font color="Maroon" title="c9">r01</font>,1)</span>)</span></span>)</span></span>)</span>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:6_2_1_1_3_1_1_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E27:6_2_1_1"><span class="lab"><font color="Green" title="E31">A24</font></span></a>, <a class="txt" href="#E10:6_2_1_1"><span class="lab"><font color="Green" title="E27">A20</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>
<a NAME="E17:6_2_1_1_3"/>
<font color="Maroon" title="c15">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="square_1.html#K3" title="SQUARE_1:func.3">sqrt</a> <font color="Maroon" title="c18">aa1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:6"><span class="lab"><font color="Green" title="E11">A4</font></span></a>, <a class="txt" href="#E5:6_2_1_1_3"><span class="lab"><font color="Green" title="E34">A27</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E18:6_2_1_1_3"/>
<font color="Maroon" title="c15">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">r</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p3">(<span class="default"><a href="xxreal_0.html#K3" title="XXREAL_0:func.3">min</a> (<font color="Maroon" title="c9">r01</font>,1)</span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c7">r</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p3">(<span class="default"><a href="xxreal_0.html#K3" title="XXREAL_0:func.3">min</a> (<font color="Maroon" title="c9">r01</font>,1)</span>)</span></span>)</span></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:6_2_1_1_3"><span class="lab"><font color="Green" title="E37">A30</font></span></a>, <a class="txt" href="#E16:6_2_1_1_3"><span class="lab"><font color="Green" title="E39">A32</font></span></a>, <a class="ref" href="xxreal_1.html#T4" onmouseover="rs('xxreal_1/T4')" onmouseout="rh()">XXREAL_1:4</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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="E29:6_2_1_1"/>
 ex <font color="Olive" title="b1">W</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font> st <br/>( <font color="Maroon" title="c5">p</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Olive" title="b1">W</font> &amp; <font color="Olive" title="b1">W</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  &amp; <font color="Maroon" title="c4">g0</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Olive" title="b1">W</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c6">V</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:6_2"><span class="lab"><font color="Green" title="E16">A9</font></span></a>, <a class="txt" href="#E8:6_2_1_1"><span class="lab"><font color="Green" title="E25">A18</font></span></a>, <a class="ref" href="xboole_1.html#T1" onmouseover="rs('xboole_1/T1')" onmouseout="rh()">XBOOLE_1:1</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="6_2_1_2"><span class="kw">suppose </span></a><a NAME="E1:6_2_1_2"/><span class="lab"><font color="Green" title="E22">A33</font></span>: 
<font color="Maroon" title="c7">r</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><a href="xxreal_0.html#K3" title="XXREAL_0:func.3">min</a> (<font color="Maroon" title="c9">r01</font>,1)</span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> 
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">W</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font> st <br/>( <font color="Maroon" title="c5">p</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Olive" title="b1">W</font> &amp; <font color="Olive" title="b1">W</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  &amp; <font color="Maroon" title="c4">g0</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Olive" title="b1">W</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c6">V</font> )</span><br/><div class="add"><span class="kw">set </span><font color="Maroon" title="c11">r4</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c7">r</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p3">(<span class="default"><a href="xxreal_0.html#K3" title="XXREAL_0:func.3">min</a> (<font color="Maroon" title="c9">r01</font>,1)</span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="xxreal_0.html#K3" title="XXREAL_0:func.3">min</a> (<font color="Maroon" title="c9">r01</font>,1)</span>)</span> <a href="square_1.html#K1" title="SQUARE_1:func.1">^2</a></span>)</span></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 3;<br/><span class="kw">reconsider </span><font color="Maroon" title="c12">G1</font> = <span class="p1"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">r1</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default">2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c7">r</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p0">(<span class="default"><a href="xxreal_0.html#K3" title="XXREAL_0:func.3">min</a> (<font color="Maroon" title="c9">r01</font>,1)</span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="xxreal_0.html#K3" title="XXREAL_0:func.3">min</a> (<font color="Maroon" title="c9">r01</font>,1)</span>)</span> <a href="square_1.html#K1" title="SQUARE_1:func.1">^2</a></span>)</span></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 3</span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c8">r1</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default">2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c7">r</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p0">(<span class="default"><a href="xxreal_0.html#K3" title="XXREAL_0:func.3">min</a> (<font color="Maroon" title="c9">r01</font>,1)</span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="xxreal_0.html#K3" title="XXREAL_0:func.3">min</a> (<font color="Maroon" title="c9">r01</font>,1)</span>)</span> <a href="square_1.html#K1" title="SQUARE_1:func.1">^2</a></span>)</span></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 3</span>)</span></span>)</span></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <a href="topmetr.html#K3" title="TOPMETR:func.3">R^1</a> <span class="kw">by</span> <span class="lab"><a class="ref" href="topmetr.html#T17" onmouseover="rs('topmetr/T17')" onmouseout="rh()">TOPMETR:17</a></span>;<br/><a NAME="E3:6_2_1_2"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c7">r</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p3">(<span class="default"><a href="xxreal_0.html#K3" title="XXREAL_0:func.3">min</a> (<font color="Maroon" title="c9">r01</font>,1)</span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="xxreal_0.html#K3" title="XXREAL_0:func.3">min</a> (<font color="Maroon" title="c9">r01</font>,1)</span>)</span> <a href="square_1.html#K1" title="SQUARE_1:func.1">^2</a></span>)</span></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 3 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:6_2"><span class="lab"><font color="Green" title="E21">A14</font></span></a>, <a class="ref" href="xreal_1.html#T139" onmouseover="rs('xreal_1/T139')" onmouseout="rh()">XREAL_1:139</a></span>;<br/><a NAME="E4:6_2_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E23">A34</font></span>: 
<font color="Maroon" title="c8">r1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c8">r1</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</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">2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c7">r</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p4">(<span class="default"><a href="xxreal_0.html#K3" title="XXREAL_0:func.3">min</a> (<font color="Maroon" title="c9">r01</font>,1)</span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="xxreal_0.html#K3" title="XXREAL_0:func.3">min</a> (<font color="Maroon" title="c9">r01</font>,1)</span>)</span> <a href="square_1.html#K1" title="SQUARE_1:func.1">^2</a></span>)</span></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 3</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T29" onmouseover="rs('xreal_1/T29')" onmouseout="rh()">XREAL_1:29</a></span>;<br/><a NAME="E5:6_2_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c8">r1</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</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">2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c7">r</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p4">(<span class="default"><a href="xxreal_0.html#K3" title="XXREAL_0:func.3">min</a> (<font color="Maroon" title="c9">r01</font>,1)</span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="xxreal_0.html#K3" title="XXREAL_0:func.3">min</a> (<font color="Maroon" title="c9">r01</font>,1)</span>)</span> <a href="square_1.html#K1" title="SQUARE_1:func.1">^2</a></span>)</span></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 3</span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c8">r1</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T19" onmouseover="rs('xreal_1/T19')" onmouseout="rh()">XREAL_1:19</a></span>;<br/><a NAME="E6:6_2_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E24">A35</font></span>: 
<font color="Maroon" title="c2">f1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">p</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c12">G1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:6_2_1_2"><span class="lab"><font color="Green" title="E23">A34</font></span></a>, <a class="ref" href="xxreal_1.html#T4" onmouseover="rs('xxreal_1/T4')" onmouseout="rh()">XXREAL_1:4</a></span>;<br/><a NAME="E7:6_2_1_2"/>
<font color="Maroon" title="c12">G1</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="jordan6.html#T35" onmouseover="rs('jordan6/T35')" onmouseout="rh()">JORDAN6:35</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c13">W1</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font><span class="kw"> such that </span><br/><a NAME="E9:6_2_1_2"/><span class="lab"><font color="Green" title="E25">A36</font></span>: 
( <font color="Maroon" title="c5">p</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c13">W1</font> &amp; <font color="Maroon" title="c13">W1</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  )
 <span class="kw">and </span><br/><a NAME="E10:6_2_1_2"/><span class="lab"><font color="Green" title="E26">A37</font></span>: 
<font color="Maroon" title="c2">f1</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c13">W1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c12">G1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:6"><span class="lab"><font color="Green" title="E8">A1</font></span></a>, <a class="txt" href="#E6:6_2_1_2"><span class="lab"><font color="Green" title="E24">A35</font></span></a>, <a class="ref" href="jgraph_2.html#T10" onmouseover="rs('jgraph_2/T10')" onmouseout="rh()">JGRAPH_2:10</a></span>;<br/><span class="kw">set </span><font color="Maroon" title="c14">W</font> = <font color="Maroon" title="c13">W1</font>;<br/><a NAME="E11:6_2_1_2"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c7">r</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p3">(<span class="default"><a href="xxreal_0.html#K3" title="XXREAL_0:func.3">min</a> (<font color="Maroon" title="c9">r01</font>,1)</span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="xxreal_0.html#K3" title="XXREAL_0:func.3">min</a> (<font color="Maroon" title="c9">r01</font>,1)</span>)</span> <a href="square_1.html#K1" title="SQUARE_1:func.1">^2</a></span>)</span></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 3 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c7">r</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><a href="xxreal_0.html#K3" title="XXREAL_0:func.3">min</a> (<font color="Maroon" title="c9">r01</font>,1)</span>)</span></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="xxreal_0.html#K3" title="XXREAL_0:func.3">min</a> (<font color="Maroon" title="c9">r01</font>,1)</span>)</span> <a href="square_1.html#K1" title="SQUARE_1:func.1">^2</a></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:6_2"><span class="lab"><font color="Green" title="E21">A14</font></span></a>, <a class="ref" href="xreal_1.html#T221" onmouseover="rs('xreal_1/T221')" onmouseout="rh()">XREAL_1:221</a></span>;<br/><a NAME="E12:6_2_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c8">r1</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</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">2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c7">r</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p4">(<span class="default"><a href="xxreal_0.html#K3" title="XXREAL_0:func.3">min</a> (<font color="Maroon" title="c9">r01</font>,1)</span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="xxreal_0.html#K3" title="XXREAL_0:func.3">min</a> (<font color="Maroon" title="c9">r01</font>,1)</span>)</span> <a href="square_1.html#K1" title="SQUARE_1:func.1">^2</a></span>)</span></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 3</span>)</span> <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="square_1.html#K1" title="SQUARE_1:func.1">^2</a></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"><span class="p3">(<span class="default">2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c7">r</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p3">(<span class="default"><a href="xxreal_0.html#K3" title="XXREAL_0:func.3">min</a> (<font color="Maroon" title="c9">r01</font>,1)</span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="xxreal_0.html#K3" title="XXREAL_0:func.3">min</a> (<font color="Maroon" title="c9">r01</font>,1)</span>)</span> <a href="square_1.html#K1" title="SQUARE_1:func.1">^2</a></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:6_2"><span class="lab"><font color="Green" title="E19">A12</font></span></a>, <a class="ref" href="xreal_1.html#T8" onmouseover="rs('xreal_1/T8')" onmouseout="rh()">XREAL_1:8</a></span>;<br/><a NAME="E13:6_2_1_2"/><span class="kw">then </span>
 <a href="square_1.html#K3" title="SQUARE_1:func.3">sqrt</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">r1</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default">2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c7">r</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p5">(<span class="default"><a href="xxreal_0.html#K3" title="XXREAL_0:func.3">min</a> (<font color="Maroon" title="c9">r01</font>,1)</span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="xxreal_0.html#K3" title="XXREAL_0:func.3">min</a> (<font color="Maroon" title="c9">r01</font>,1)</span>)</span> <a href="square_1.html#K1" title="SQUARE_1:func.1">^2</a></span>)</span></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 3</span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="square_1.html#K3" title="SQUARE_1:func.3">sqrt</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">r</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p3">(<span class="default"><a href="xxreal_0.html#K3" title="XXREAL_0:func.3">min</a> (<font color="Maroon" title="c9">r01</font>,1)</span>)</span></span>)</span> <a href="square_1.html#K1" title="SQUARE_1:func.1">^2</a></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:6_2"><span class="lab"><font color="Green" title="E17">A10</font></span></a>, <a class="txt" href="#E16:6_2"><span class="lab"><font color="Green" title="E20">A13</font></span></a>, <a class="txt" href="#E8:6_2"><span class="lab"><font color="Green" title="E15">A8</font></span></a>, <a class="ref" href="square_1.html#T26" onmouseover="rs('square_1/T26')" onmouseout="rh()">SQUARE_1:26</a></span>;<br/><a NAME="E14:6_2_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E27">A38</font></span>: 
<font color="Maroon" title="c7">r</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><a href="xxreal_0.html#K3" title="XXREAL_0:func.3">min</a> (<font color="Maroon" title="c9">r01</font>,1)</span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="square_1.html#K3" title="SQUARE_1:func.3">sqrt</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">r1</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default">2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c7">r</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p5">(<span class="default"><a href="xxreal_0.html#K3" title="XXREAL_0:func.3">min</a> (<font color="Maroon" title="c9">r01</font>,1)</span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="xxreal_0.html#K3" title="XXREAL_0:func.3">min</a> (<font color="Maroon" title="c9">r01</font>,1)</span>)</span> <a href="square_1.html#K1" title="SQUARE_1:func.1">^2</a></span>)</span></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 3</span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:6_2"><span class="lab"><font color="Green" title="E20">A13</font></span></a>, <a class="txt" href="#E7:6_2"><span class="lab"><font color="Green" title="E14">A7</font></span></a>, <a class="ref" href="square_1.html#T22" onmouseover="rs('square_1/T22')" onmouseout="rh()">SQUARE_1:22</a></span>;<br/><a NAME="E15:6_2_1_2"/>
<font color="Maroon" title="c4">g0</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c13">W1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">r</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p3">(<span class="default"><a href="xxreal_0.html#K3" title="XXREAL_0:func.3">min</a> (<font color="Maroon" title="c9">r01</font>,1)</span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c7">r</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p3">(<span class="default"><a href="xxreal_0.html#K3" title="XXREAL_0:func.3">min</a> (<font color="Maroon" title="c9">r01</font>,1)</span>)</span></span>)</span></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span>
 <div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="6_2_1_2_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c15">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="tarski.html#D3" onmouseover="rs('tarski/D3')" onmouseout="rh()">TARSKI:def 3</a> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c15">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">g0</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c13">W1</font> or <font color="Maroon" title="c15">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">r</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p3">(<span class="default"><a href="xxreal_0.html#K3" title="XXREAL_0:func.3">min</a> (<font color="Maroon" title="c9">r01</font>,1)</span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c7">r</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p3">(<span class="default"><a href="xxreal_0.html#K3" title="XXREAL_0:func.3">min</a> (<font color="Maroon" title="c9">r01</font>,1)</span>)</span></span>)</span></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span> )</span><br/>

<span class="kw">assume </span><a NAME="E1:6_2_1_2_1"/>
<font color="Maroon" title="c15">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">g0</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c13">W1</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c15">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">r</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p3">(<span class="default"><a href="xxreal_0.html#K3" title="XXREAL_0:func.3">min</a> (<font color="Maroon" title="c9">r01</font>,1)</span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c7">r</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p3">(<span class="default"><a href="xxreal_0.html#K3" title="XXREAL_0:func.3">min</a> (<font color="Maroon" title="c9">r01</font>,1)</span>)</span></span>)</span></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span></span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c16">z</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> <span class="kw"> such that </span><br/><a NAME="E3:6_2_1_2_1"/><span class="lab"><font color="Green" title="E28">A39</font></span>: 
<font color="Maroon" title="c16">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c4">g0</font>
 <span class="kw">and </span><br/><a NAME="E4:6_2_1_2_1"/><span class="lab"><font color="Green" title="E29">A40</font></span>: 
<font color="Maroon" title="c16">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c13">W1</font>
 <span class="kw">and </span><br/><a NAME="E5:6_2_1_2_1"/><span class="lab"><font color="Green" title="E30">A41</font></span>: 
<font color="Maroon" title="c4">g0</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">x</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#D6" onmouseover="rs('funct_1/D6')" onmouseout="rh()">FUNCT_1:def 6</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c17">pz</font> = <font color="Maroon" title="c16">z</font> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">X</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:6_2_1_2_1"><span class="lab"><font color="Green" title="E28">A39</font></span></a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c18">aa1</font> = <font color="Maroon" title="c2">f1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c17">pz</font> as   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a> ;<br/>
<a NAME="E8:6_2_1_2_1"/><span class="lab"><font color="Green" title="E31">A42</font></span>: 
 ex <font color="Olive" title="b1">r9</font> being   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a> st <br/>( <font color="Maroon" title="c2">f1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c17">pz</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">r9</font> &amp; <font color="Olive" title="b1">r9</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:6"><span class="lab"><font color="Green" title="E9">A2</font></span></a></span>;<br/>
<a NAME="E9:6_2_1_2_1"/>
<font color="Maroon" title="c17">pz</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">X</font>
 
;<br/>
<a NAME="E10:6_2_1_2_1"/><span class="kw">then </span>
<font color="Maroon" title="c17">pz</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c2">f1</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D1" onmouseover="rs('funct_2/D1')" onmouseout="rh()">FUNCT_2:def 1</a></span>;<br/>
<a NAME="E11:6_2_1_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E32">A43</font></span>: 
<font color="Maroon" title="c2">f1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c17">pz</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c2">f1</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c13">W1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:6_2_1_2_1"><span class="lab"><font color="Green" title="E29">A40</font></span></a>, <a class="ref" href="funct_1.html#D6" onmouseover="rs('funct_1/D6')" onmouseout="rh()">FUNCT_1:def 6</a></span>;<br/>
<a NAME="E12:6_2_1_2_1"/><span class="kw">then </span>
<font color="Maroon" title="c18">aa1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c8">r1</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</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">2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c7">r</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p4">(<span class="default"><a href="xxreal_0.html#K3" title="XXREAL_0:func.3">min</a> (<font color="Maroon" title="c9">r01</font>,1)</span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="xxreal_0.html#K3" title="XXREAL_0:func.3">min</a> (<font color="Maroon" title="c9">r01</font>,1)</span>)</span> <a href="square_1.html#K1" title="SQUARE_1:func.1">^2</a></span>)</span></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 3</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:6_2_1_2"><span class="lab"><font color="Green" title="E26">A37</font></span></a>, <a class="ref" href="xxreal_1.html#T4" onmouseover="rs('xxreal_1/T4')" onmouseout="rh()">XXREAL_1:4</a></span>;<br/>
<a NAME="E13:6_2_1_2_1"/><span class="kw">then </span>
 <a href="square_1.html#K3" title="SQUARE_1:func.3">sqrt</a> <font color="Maroon" title="c18">aa1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="square_1.html#K3" title="SQUARE_1:func.3">sqrt</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">r1</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default">2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c7">r</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p5">(<span class="default"><a href="xxreal_0.html#K3" title="XXREAL_0:func.3">min</a> (<font color="Maroon" title="c9">r01</font>,1)</span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="xxreal_0.html#K3" title="XXREAL_0:func.3">min</a> (<font color="Maroon" title="c9">r01</font>,1)</span>)</span> <a href="square_1.html#K1" title="SQUARE_1:func.1">^2</a></span>)</span></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 3</span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:6_2_1_2_1"><span class="lab"><font color="Green" title="E31">A42</font></span></a>, <a class="ref" href="square_1.html#T27" onmouseover="rs('square_1/T27')" onmouseout="rh()">SQUARE_1:27</a></span>;<br/>
<a NAME="E14:6_2_1_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E33">A44</font></span>: 
 <a href="square_1.html#K3" title="SQUARE_1:func.3">sqrt</a> <font color="Maroon" title="c18">aa1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c7">r</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><a href="xxreal_0.html#K3" title="XXREAL_0:func.3">min</a> (<font color="Maroon" title="c9">r01</font>,1)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:6_2_1_2"><span class="lab"><font color="Green" title="E27">A38</font></span></a>, <a class="ref" href="xxreal_0.html#T2" onmouseover="rs('xxreal_0/T2')" onmouseout="rh()">XXREAL_0:2</a></span>;<br/>
<a NAME="E15:6_2_1_2_1"/><span class="lab"><font color="Green" title="E34">A45</font></span>: 
<font color="Maroon" title="c8">r1</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</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">2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c7">r</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p4">(<span class="default"><a href="xxreal_0.html#K3" title="XXREAL_0:func.3">min</a> (<font color="Maroon" title="c9">r01</font>,1)</span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="xxreal_0.html#K3" title="XXREAL_0:func.3">min</a> (<font color="Maroon" title="c9">r01</font>,1)</span>)</span> <a href="square_1.html#K1" title="SQUARE_1:func.1">^2</a></span>)</span></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 3</span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c18">aa1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:6_2_1_2"><span class="lab"><font color="Green" title="E26">A37</font></span></a>, <a class="txt" href="#E11:6_2_1_2_1"><span class="lab"><font color="Green" title="E32">A43</font></span></a>, <a class="ref" href="xxreal_1.html#T4" onmouseover="rs('xxreal_1/T4')" onmouseout="rh()">XXREAL_1:4</a></span>;<br/>
<div><span class="lab"><font color="Green" title="E35">A46</font></span>: <a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="6_2_1_2_1_1"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( ( <font color="Maroon" title="c7">r</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><a href="xxreal_0.html#K3" title="XXREAL_0:func.3">min</a> (<font color="Maroon" title="c9">r01</font>,1)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  &amp; <font color="Maroon" title="c7">r</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><a href="xxreal_0.html#K3" title="XXREAL_0:func.3">min</a> (<font color="Maroon" title="c9">r01</font>,1)</span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="square_1.html#K3" title="SQUARE_1:func.3">sqrt</a> <font color="Maroon" title="c18">aa1</font> ) or ( <font color="Maroon" title="c7">r</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><a href="xxreal_0.html#K3" title="XXREAL_0:func.3">min</a> (<font color="Maroon" title="c9">r01</font>,1)</span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  &amp; <font color="Maroon" title="c7">r</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><a href="xxreal_0.html#K3" title="XXREAL_0:func.3">min</a> (<font color="Maroon" title="c9">r01</font>,1)</span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="square_1.html#K3" title="SQUARE_1:func.3">sqrt</a> <font color="Maroon" title="c18">aa1</font> ) )</span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="6_2_1_2_1_1_1"><span class="kw">per </span></a><a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:6_2_1_2_1_1_1"/>
( <font color="Maroon" title="c7">r</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><a href="xxreal_0.html#K3" title="XXREAL_0:func.3">min</a> (<font color="Maroon" title="c9">r01</font>,1)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  or <font color="Maroon" title="c7">r</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><a href="xxreal_0.html#K3" title="XXREAL_0:func.3">min</a> (<font color="Maroon" title="c9">r01</font>,1)</span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:6_2_1_2"><span class="lab"><font color="Green" title="E22">A33</font></span></a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="6_2_1_2_1_1_1_1"><span class="kw">case </span></a><a NAME="E1:6_2_1_2_1_1_1_1"/>
<font color="Maroon" title="c7">r</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><a href="xxreal_0.html#K3" title="XXREAL_0:func.3">min</a> (<font color="Maroon" title="c9">r01</font>,1)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> 
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c7">r</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><a href="xxreal_0.html#K3" title="XXREAL_0:func.3">min</a> (<font color="Maroon" title="c9">r01</font>,1)</span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="square_1.html#K3" title="SQUARE_1:func.3">sqrt</a> <font color="Maroon" title="c18">aa1</font></span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:6_2_1_2_1_1_1_1"/>
<font color="Maroon" title="c7">r</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><a href="xxreal_0.html#K3" title="XXREAL_0:func.3">min</a> (<font color="Maroon" title="c9">r01</font>,1)</span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="square_1.html#K3" title="SQUARE_1:func.3">sqrt</a> <font color="Maroon" title="c18">aa1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:6_2"><span class="lab"><font color="Green" title="E19">A12</font></span></a>, <a class="txt" href="#E15:6_2_1_2_1"><span class="lab"><font color="Green" title="E34">A45</font></span></a>, <a class="ref" href="square_1.html#T17" onmouseover="rs('square_1/T17')" onmouseout="rh()">SQUARE_1:17</a>, <a class="ref" href="square_1.html#T27" onmouseover="rs('square_1/T27')" onmouseout="rh()">SQUARE_1:27</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="6_2_1_2_1_1_1_2"><span class="kw">case </span></a><a NAME="E1:6_2_1_2_1_1_1_2"/>
<font color="Maroon" title="c7">r</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><a href="xxreal_0.html#K3" title="XXREAL_0:func.3">min</a> (<font color="Maroon" title="c9">r01</font>,1)</span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> 
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c7">r</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><a href="xxreal_0.html#K3" title="XXREAL_0:func.3">min</a> (<font color="Maroon" title="c9">r01</font>,1)</span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="square_1.html#K3" title="SQUARE_1:func.3">sqrt</a> <font color="Maroon" title="c18">aa1</font></span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:6_2_1_2_1_1_1_2"/>
<font color="Maroon" title="c7">r</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><a href="xxreal_0.html#K3" title="XXREAL_0:func.3">min</a> (<font color="Maroon" title="c9">r01</font>,1)</span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="square_1.html#K3" title="SQUARE_1:func.3">sqrt</a> <font color="Maroon" title="c18">aa1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:6_2_1_2_1"><span class="lab"><font color="Green" title="E31">A42</font></span></a>, <a class="ref" href="square_1.html#T17" onmouseover="rs('square_1/T17')" onmouseout="rh()">SQUARE_1:17</a>, <a class="ref" href="square_1.html#T26" onmouseover="rs('square_1/T26')" onmouseout="rh()">SQUARE_1:26</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>
<a NAME="E17:6_2_1_2_1"/>
<font color="Maroon" title="c15">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="square_1.html#K3" title="SQUARE_1:func.3">sqrt</a> <font color="Maroon" title="c18">aa1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:6"><span class="lab"><font color="Green" title="E11">A4</font></span></a>, <a class="txt" href="#E5:6_2_1_2_1"><span class="lab"><font color="Green" title="E30">A41</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E18:6_2_1_2_1"/>
<font color="Maroon" title="c15">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">r</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p3">(<span class="default"><a href="xxreal_0.html#K3" title="XXREAL_0:func.3">min</a> (<font color="Maroon" title="c9">r01</font>,1)</span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c7">r</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p3">(<span class="default"><a href="xxreal_0.html#K3" title="XXREAL_0:func.3">min</a> (<font color="Maroon" title="c9">r01</font>,1)</span>)</span></span>)</span></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:6_2_1_2_1"><span class="lab"><font color="Green" title="E33">A44</font></span></a>, <a class="txt" href="#E16:6_2_1_2_1"><span class="lab"><font color="Green" title="E35">A46</font></span></a>, <a class="ref" href="xxreal_1.html#T4" onmouseover="rs('xxreal_1/T4')" onmouseout="rh()">XXREAL_1:4</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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="E16:6_2_1_2"/>
 ex <font color="Olive" title="b1">W</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font> st <br/>( <font color="Maroon" title="c5">p</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Olive" title="b1">W</font> &amp; <font color="Olive" title="b1">W</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  &amp; <font color="Maroon" title="c4">g0</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Olive" title="b1">W</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c6">V</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:6_2"><span class="lab"><font color="Green" title="E16">A9</font></span></a>, <a class="txt" href="#E9:6_2_1_2"><span class="lab"><font color="Green" title="E25">A36</font></span></a>, <a class="ref" href="xboole_1.html#T1" onmouseover="rs('xboole_1/T1')" onmouseout="rh()">XBOOLE_1:1</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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></div><span class="kw">end;</span></div>

</div><span class="kw">end;</span></div>
<a NAME="E9:6"/><span class="kw">then </span><span class="lab"><font color="Green" title="E12">A47</font></span>: 
<font color="Maroon" title="c4">g0</font> is  <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="jgraph_2.html#T10" onmouseover="rs('jgraph_2/T10')" onmouseout="rh()">JGRAPH_2:10</a></span>;<br/>
<a NAME="E10:6"/>
 for <font color="Olive" title="b1">p</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">X</font><br/>  for <font color="Olive" title="b2">r11</font> being   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a>  st <font color="Maroon" title="c2">f1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">r11</font> holds <br/><font color="Maroon" title="c4">g0</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="square_1.html#K3" title="SQUARE_1:func.3">sqrt</a> <font color="Olive" title="b2">r11</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:6"><span class="lab"><font color="Green" title="E11">A4</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E11:6"/>
 ex <font color="Olive" title="b1">g</font> being   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c1">X</font>,<a href="topmetr.html#K3" title="TOPMETR:func.3">R^1</a> st <br/>( (  for <font color="Olive" title="b2">p</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">X</font><br/>  for <font color="Olive" title="b3">r1</font> being   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a>  st <font color="Maroon" title="c2">f1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">r1</font> holds <br/><font color="Olive" title="b1">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="square_1.html#K3" title="SQUARE_1:func.3">sqrt</a> <font color="Olive" title="b3">r1</font> ) &amp; <font color="Olive" title="b1">g</font> is  <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a>  )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:6"><span class="lab"><font color="Green" title="E12">A47</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
