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<span class="kw">let </span><font color="Maroon" title="c1">n</font> be   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</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">A</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">n</font></span>)</span>  st <font color="Olive" title="b1">A</font> is  <a href="rltopsp1.html#V9" title="RLTOPSP1:attr.9">bounded</a>  holds <br/> <a href="jordan2c.html#K1" title="JORDAN2C:func.1">BDD</a> <font color="Olive" title="b1">A</font> is  <a href="rltopsp1.html#V9" title="RLTOPSP1:attr.9">bounded</a> </span><br/><span class="kw">let </span><font color="Maroon" title="c2">A</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">n</font></span>)</span>; <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">A</font> is  <a href="rltopsp1.html#V9" title="RLTOPSP1:attr.9">bounded</a>  implies  <a href="jordan2c.html#K1" title="JORDAN2C:func.1">BDD</a> <font color="Maroon" title="c2">A</font> is  <a href="rltopsp1.html#V9" title="RLTOPSP1:attr.9">bounded</a>  )</span><br/>



<span class="kw">assume </span><a NAME="E1:114"/>
<font color="Maroon" title="c2">A</font> is  <a href="rltopsp1.html#V9" title="RLTOPSP1:attr.9">bounded</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">  <a href="jordan2c.html#K1" title="JORDAN2C:func.1">BDD</a> <font color="Maroon" title="c2">A</font> is  <a href="rltopsp1.html#V9" title="RLTOPSP1:attr.9">bounded</a> </span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c3">r</font> being   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a><span class="kw"> such that </span><br/><a NAME="E3:114"/><span class="lab"><font color="Green" title="E86">A1</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 <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">n</font></span>)</span>  st <font color="Olive" title="b1">q</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c2">A</font> holds <br/><span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><font color="Olive" title="b1">q</font></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">r</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="jordan2c.html#T34" target="_self" onmouseover="rs('jordan2c/T34')" onmouseout="rh()">Th21</a></span>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="114_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="E1:114_1"/>
( <font color="Maroon" title="c1">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1 or <font color="Maroon" title="c1">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="114_1_1"><span class="kw">suppose </span></a><a NAME="E1:114_1_1"/><span class="lab"><font color="Green" title="E87">A2</font></span>: 
<font color="Maroon" title="c1">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1
 ; <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="jordan2c.html#K1" title="JORDAN2C:func.1">BDD</a> <font color="Maroon" title="c2">A</font> is  <a href="rltopsp1.html#V9" title="RLTOPSP1:attr.9">bounded</a> </span><br/><div class="add"><span class="kw">set </span><font color="Maroon" title="c4">a</font> = <font color="Maroon" title="c3">r</font>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c5">P</font> = <span class="p1">(<span class="default"><a href="euclid.html#K1" title="EUCLID:func.1">REAL</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">q</font> where <font color="Olive" title="b1">q</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">n</font></span>)</span> : <span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><font color="Olive" title="b1">q</font></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">r</font> </span> } </span>  as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">n</font></span>)</span> <span class="kw">by</span> <span class="lab"><a class="ref" href="euclid.html#T22" onmouseover="rs('euclid/T22')" onmouseout="rh()">EUCLID:22</a></span>;<br/><a NAME="E3:114_1_1"/><span class="lab"><font color="Green" title="E88">A3</font></span>: 
<font color="Maroon" title="c5">P</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">A</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> 
 <div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="114_1_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c6">z</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="c6">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">P</font> or <font color="Maroon" title="c6">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  )</span><br/>

<span class="kw">assume </span><a NAME="E1:114_1_1_1"/><span class="lab"><font color="Green" title="E89">A4</font></span>: 
<font color="Maroon" title="c6">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">P</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="c6">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span><br/>

<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c7">q0</font> = <font color="Maroon" title="c6">z</font> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">n</font></span>)</span> ;<br/>
<a NAME="E3:114_1_1_1"/>
 not <font color="Maroon" title="c6">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">q</font> where <font color="Olive" title="b1">q</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">n</font></span>)</span> : <span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><font color="Olive" title="b1">q</font></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">r</font> </span> } </span> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:114_1_1_1"><span class="lab"><font color="Green" title="E89">A4</font></span></a>, <a class="ref" href="xboole_0.html#D5" onmouseover="rs('xboole_0/D5')" onmouseout="rh()">XBOOLE_0:def 5</a></span>;<br/>
<a NAME="E4:114_1_1_1"/><span class="kw">then </span>
<span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><font color="Maroon" title="c7">q0</font></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c3">r</font>
 
;<br/>
<a NAME="E5:114_1_1_1"/><span class="kw">then </span>
 not <font color="Maroon" title="c7">q0</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c2">A</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:114"><span class="lab"><font color="Green" title="E86">A1</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E6:114_1_1_1"/>
<font color="Maroon" title="c6">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D5" onmouseover="rs('xboole_0/D5')" onmouseout="rh()">XBOOLE_0:def 5</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="E4:114_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E89">A5</font></span>: 
 <a href="connsp_3.html#K4" title="CONNSP_3:func.4">Down</a> (<font color="Maroon" title="c5">P</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a></span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">P</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T28" onmouseover="rs('xboole_1/T28')" onmouseout="rh()">XBOOLE_1:28</a></span>;<br/><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="114_1_1_2"><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="jordan2c.html#K1" title="JORDAN2C:func.1">BDD</a> <font color="Maroon" title="c2">A</font> is  <a href="rltopsp1.html#V9" title="RLTOPSP1:attr.9">bounded</a> </span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="114_1_1_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="E1:114_1_1_2_1"/>
( <font color="Maroon" title="c1">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 2 or <font color="Maroon" title="c1">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 2 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="114_1_1_2_1_1"><span class="kw">suppose </span></a><a NAME="E1:114_1_1_2_1_1"/>
<font color="Maroon" title="c1">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 2
 ; <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="jordan2c.html#K1" title="JORDAN2C:func.1">BDD</a> <font color="Maroon" title="c2">A</font> is  <a href="rltopsp1.html#V9" title="RLTOPSP1:attr.9">bounded</a> </span><br/><div class="add"><a NAME="E2:114_1_1_2_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E90">A6</font></span>: 
<font color="Maroon" title="c5">P</font> is  <a href="connsp_1.html#V2" title="CONNSP_1:attr.2">connected</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="jordan2c.html#T53" target="_self" onmouseover="rs('jordan2c/T53')" onmouseout="rh()">Th40</a></span>;<br/><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="114_1_1_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">  <a href="jordan2c.html#K1" title="JORDAN2C:func.1">BDD</a> <font color="Maroon" title="c2">A</font> is  <a href="rltopsp1.html#V9" title="RLTOPSP1:attr.9">bounded</a> </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:114_1_1_2_1_1_1"/>
 not  <a href="jordan2c.html#K1" title="JORDAN2C:func.1">BDD</a> <font color="Maroon" title="c2">A</font> is  <a href="rltopsp1.html#V9" title="RLTOPSP1:attr.9">bounded</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/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c6">q</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">n</font></span>)</span><span class="kw"> such that </span><br/><a NAME="E3:114_1_1_2_1_1_1"/><span class="lab"><font color="Green" title="E91">A7</font></span>: 
<font color="Maroon" title="c6">q</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="jordan2c.html#K1" title="JORDAN2C:func.1">BDD</a> <font color="Maroon" title="c2">A</font>
 <span class="kw">and </span><br/><a NAME="E4:114_1_1_2_1_1_1"/><span class="lab"><font color="Green" title="E92">A8</font></span>: 
 not <span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><font color="Maroon" title="c6">q</font></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">r</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="jordan2c.html#T34" target="_self" onmouseover="rs('jordan2c/T34')" onmouseout="rh()">Th21</a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c7">y</font> being    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a> <span class="kw"> such that </span><br/><a NAME="E6:114_1_1_2_1_1_1"/><span class="lab"><font color="Green" title="E93">A9</font></span>: 
<font color="Maroon" title="c6">q</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c7">y</font>
 <span class="kw">and </span><br/><a NAME="E7:114_1_1_2_1_1_1"/><span class="lab"><font color="Green" title="E94">A10</font></span>: 
<font color="Maroon" title="c7">y</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">B3</font> where <font color="Olive" title="b1">B3</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">n</font></span>)</span> : <font color="Olive" title="b1">B3</font> <a href="jordan2c.html#R1" title="JORDAN2C:pred.1">is_inside_component_of</a> <font color="Maroon" title="c2">A</font> </span> } </span> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:114_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E91">A7</font></span></a>, <a class="ref" href="tarski.html#D4" onmouseover="rs('tarski/D4')" onmouseout="rh()">TARSKI:def 4</a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c8">B3</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">n</font></span>)</span><span class="kw"> such that </span><br/><a NAME="E9:114_1_1_2_1_1_1"/><span class="lab"><font color="Green" title="E95">A11</font></span>: 
<font color="Maroon" title="c7">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">B3</font>
 <span class="kw">and </span><br/><a NAME="E10:114_1_1_2_1_1_1"/><span class="lab"><font color="Green" title="E96">A12</font></span>: 
<font color="Maroon" title="c8">B3</font> <a href="jordan2c.html#R1" title="JORDAN2C:pred.1">is_inside_component_of</a> <font color="Maroon" title="c2">A</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:114_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E94">A10</font></span></a></span>;<br/><a NAME="E11:114_1_1_2_1_1_1"/>
<font color="Maroon" title="c6">q</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 <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">n</font></span>)</span>
 ;<br/><a NAME="E12:114_1_1_2_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E97">A13</font></span>: 
<font color="Maroon" title="c6">q</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="euclid.html#K1" title="EUCLID:func.1">REAL</a> <font color="Maroon" title="c1">n</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="euclid.html#T22" onmouseover="rs('euclid/T22')" onmouseout="rh()">EUCLID:22</a></span>;<br/><a NAME="E13:114_1_1_2_1_1_1"/>
<font color="Maroon" title="c8">B3</font> <a href="connsp_1.html#R3" title="CONNSP_1:pred.3">is_a_component_of</a> <font color="Maroon" title="c2">A</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:114_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E96">A12</font></span></a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c9">B4</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a></span>)</span></span>)</span><span class="kw"> such that </span><br/><a NAME="E15:114_1_1_2_1_1_1"/><span class="lab"><font color="Green" title="E98">A14</font></span>: 
<font color="Maroon" title="c9">B4</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">B3</font>
 <span class="kw">and </span><br/><a NAME="E16:114_1_1_2_1_1_1"/><span class="lab"><font color="Green" title="E99">A15</font></span>: 
<font color="Maroon" title="c9">B4</font> is  <a href="connsp_1.html#V3" title="CONNSP_1:attr.3">a_component</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="connsp_1.html#D6" onmouseover="rs('connsp_1/D6')" onmouseout="rh()">CONNSP_1:def 6</a></span>;<br/><a NAME="E17:114_1_1_2_1_1_1"/>
 for <font color="Olive" title="b1">q2</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">n</font></span>)</span>  st <font color="Olive" title="b1">q2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">q</font> holds <br/><span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><font color="Olive" title="b1">q2</font></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c3">r</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:114_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E92">A8</font></span></a></span>;<br/><a NAME="E18:114_1_1_2_1_1_1"/><span class="kw">then </span>
 not <font color="Maroon" title="c6">q</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">q2</font> where <font color="Olive" title="b1">q2</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">n</font></span>)</span> : <span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><font color="Olive" title="b1">q2</font></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">r</font> </span> } </span> 
 ;<br/><a NAME="E19:114_1_1_2_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c6">q</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c5">P</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:114_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E97">A13</font></span></a>, <a class="ref" href="xboole_0.html#D5" onmouseover="rs('xboole_0/D5')" onmouseout="rh()">XBOOLE_0:def 5</a></span>;<br/><a NAME="E20:114_1_1_2_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c5">P</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c9">B4</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="struct_0.html#K1" title="STRUCT_0:func.1">{}</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:114_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E93">A9</font></span></a>, <a class="txt" href="#E9:114_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E95">A11</font></span></a>, <a class="txt" href="#E15:114_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E98">A14</font></span></a>, <a class="ref" href="xboole_0.html#D4" onmouseover="rs('xboole_0/D4')" onmouseout="rh()">XBOOLE_0:def 4</a></span>;<br/><a NAME="E21:114_1_1_2_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c5">P</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c9">B4</font>
 ;<br/><a NAME="E22:114_1_1_2_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E100">A16</font></span>: 
<font color="Maroon" title="c5">P</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c9">B4</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:114_1_1"><span class="lab"><font color="Green" title="E89">A5</font></span></a>, <a class="txt" href="#E2:114_1_1_2_1_1"><span class="lab"><font color="Green" title="E90">A6</font></span></a>, <a class="txt" href="#E16:114_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E99">A15</font></span></a>, <a class="ref" href="connsp_1.html#T36" onmouseover="rs('connsp_1/T36')" onmouseout="rh()">CONNSP_1:36</a>, <a class="ref" href="connsp_1.html#T46" onmouseover="rs('connsp_1/T46')" onmouseout="rh()">CONNSP_1:46</a></span>;<br/><a NAME="E23:114_1_1_2_1_1_1"/>
<font color="Maroon" title="c8">B3</font> is  <a href="rltopsp1.html#V9" title="RLTOPSP1:attr.9">bounded</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:114_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E96">A12</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E24:114_1_1_2_1_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:114_1_1"><span class="lab"><font color="Green" title="E87">A2</font></span></a>, <a class="txt" href="#E15:114_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E98">A14</font></span></a>, <a class="txt" href="#E22:114_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E100">A16</font></span></a>, <a class="ref" href="jordan2c.html#T54" target="_self" onmouseover="rs('jordan2c/T54')" onmouseout="rh()">Th41</a>, <a class="ref" href="rltopsp1.html#T42" onmouseover="rs('rltopsp1/T42')" onmouseout="rh()">RLTOPSP1:42</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="E4:114_1_1_2_1_1"/>
 <a href="jordan2c.html#K1" title="JORDAN2C:func.1">BDD</a> <font color="Maroon" title="c2">A</font> is  <a href="rltopsp1.html#V9" title="RLTOPSP1:attr.9">bounded</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"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="114_1_1_2_1_2"><span class="kw">suppose </span></a><a NAME="E1:114_1_1_2_1_2"/><span class="lab"><font color="Green" title="E90">A17</font></span>: 
<font color="Maroon" title="c1">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 2
 ; <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="jordan2c.html#K1" title="JORDAN2C:func.1">BDD</a> <font color="Maroon" title="c2">A</font> is  <a href="rltopsp1.html#V9" title="RLTOPSP1:attr.9">bounded</a> </span><br/><div class="add"><a NAME="E2:114_1_1_2_1_2"/>
<span class="p1"> { <span class="default"> <font color="Olive" title="b1">q</font> where <font color="Olive" title="b1">q</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">n</font></span>)</span> :  for <font color="Olive" title="b2">r2</font> being   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a>  st <font color="Olive" title="b1">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="jordan2b.html#K1" title="JORDAN2B:func.1">|[</a><span class="default"><font color="Olive" title="b2">r2</font></span><a href="jordan2b.html#K1" title="JORDAN2B:func.1">]|</a></span> holds <br/><font color="Olive" title="b2">r2</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c3">r</font> </span> } </span>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">n</font></span>)</span>
 <div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="114_1_1_2_1_2_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c6">z</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="c6">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">q</font> where <font color="Olive" title="b1">q</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">n</font></span>)</span> :  for <font color="Olive" title="b2">r2</font> being   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a>  st <font color="Olive" title="b1">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="jordan2b.html#K1" title="JORDAN2B:func.1">|[</a><span class="default"><font color="Olive" title="b2">r2</font></span><a href="jordan2b.html#K1" title="JORDAN2B:func.1">]|</a></span> holds <br/><font color="Olive" title="b2">r2</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c3">r</font> </span> } </span>  or <font color="Maroon" title="c6">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">n</font></span>)</span> )</span><br/>

<span class="kw">assume </span><a NAME="E1:114_1_1_2_1_2_1"/>
<font color="Maroon" title="c6">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">q</font> where <font color="Olive" title="b1">q</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">n</font></span>)</span> :  for <font color="Olive" title="b2">r2</font> being   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a>  st <font color="Olive" title="b1">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="jordan2b.html#K1" title="JORDAN2B:func.1">|[</a><span class="default"><font color="Olive" title="b2">r2</font></span><a href="jordan2b.html#K1" title="JORDAN2B:func.1">]|</a></span> holds <br/><font color="Olive" title="b2">r2</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c3">r</font> </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="c6">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">n</font></span>)</span></span><br/>

<a NAME="E2:114_1_1_2_1_2_1"/><span class="kw">then </span>
 ex <font color="Olive" title="b1">q</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">n</font></span>)</span> st <br/>( <font color="Olive" title="b1">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">z</font> &amp; (  for <font color="Olive" title="b2">r2</font> being   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a>  st <font color="Olive" title="b1">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="jordan2b.html#K1" title="JORDAN2B:func.1">|[</a><span class="default"><font color="Olive" title="b2">r2</font></span><a href="jordan2b.html#K1" title="JORDAN2B:func.1">]|</a></span> holds <br/><font color="Olive" title="b2">r2</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c3">r</font> ) )
 
;<br/>
<span class="kw">hence </span><a NAME="E3:114_1_1_2_1_2_1"/>
<font color="Maroon" title="c6">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">n</font></span>)</span>
 ; <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">then </span><span class="kw">reconsider </span><font color="Maroon" title="c6">P2</font> = <span class="p1"> { <span class="default"> <font color="Olive" title="b1">q</font> where <font color="Olive" title="b1">q</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">n</font></span>)</span> :  for <font color="Olive" title="b2">r2</font> being   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a>  st <font color="Olive" title="b1">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="jordan2b.html#K1" title="JORDAN2B:func.1">|[</a><span class="default"><font color="Olive" title="b2">r2</font></span><a href="jordan2b.html#K1" title="JORDAN2B:func.1">]|</a></span> holds <br/><font color="Olive" title="b2">r2</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c3">r</font> </span> } </span>  as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">n</font></span>)</span> ;<br/><a NAME="E4:114_1_1_2_1_2"/>
<span class="p1"> { <span class="default"> <font color="Olive" title="b1">q</font> where <font color="Olive" title="b1">q</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">n</font></span>)</span> :  for <font color="Olive" title="b2">r2</font> being   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a>  st <font color="Olive" title="b1">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="jordan2b.html#K1" title="JORDAN2B:func.1">|[</a><span class="default"><font color="Olive" title="b2">r2</font></span><a href="jordan2b.html#K1" title="JORDAN2B:func.1">]|</a></span> holds <br/><font color="Olive" title="b2">r2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <font color="Maroon" title="c3">r</font> </span> } </span>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">n</font></span>)</span>
 <div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="114_1_1_2_1_2_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c7">z</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="c7">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">q</font> where <font color="Olive" title="b1">q</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">n</font></span>)</span> :  for <font color="Olive" title="b2">r2</font> being   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a>  st <font color="Olive" title="b1">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="jordan2b.html#K1" title="JORDAN2B:func.1">|[</a><span class="default"><font color="Olive" title="b2">r2</font></span><a href="jordan2b.html#K1" title="JORDAN2B:func.1">]|</a></span> holds <br/><font color="Olive" title="b2">r2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <font color="Maroon" title="c3">r</font> </span> } </span>  or <font color="Maroon" title="c7">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">n</font></span>)</span> )</span><br/>

<span class="kw">assume </span><a NAME="E1:114_1_1_2_1_2_2"/>
<font color="Maroon" title="c7">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">q</font> where <font color="Olive" title="b1">q</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">n</font></span>)</span> :  for <font color="Olive" title="b2">r2</font> being   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a>  st <font color="Olive" title="b1">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="jordan2b.html#K1" title="JORDAN2B:func.1">|[</a><span class="default"><font color="Olive" title="b2">r2</font></span><a href="jordan2b.html#K1" title="JORDAN2B:func.1">]|</a></span> holds <br/><font color="Olive" title="b2">r2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <font color="Maroon" title="c3">r</font> </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">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">n</font></span>)</span></span><br/>

<a NAME="E2:114_1_1_2_1_2_2"/><span class="kw">then </span>
 ex <font color="Olive" title="b1">q</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">n</font></span>)</span> st <br/>( <font color="Olive" title="b1">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">z</font> &amp; (  for <font color="Olive" title="b2">r2</font> being   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a>  st <font color="Olive" title="b1">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="jordan2b.html#K1" title="JORDAN2B:func.1">|[</a><span class="default"><font color="Olive" title="b2">r2</font></span><a href="jordan2b.html#K1" title="JORDAN2B:func.1">]|</a></span> holds <br/><font color="Olive" title="b2">r2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <font color="Maroon" title="c3">r</font> ) )
 
;<br/>
<span class="kw">hence </span><a NAME="E3:114_1_1_2_1_2_2"/>
<font color="Maroon" title="c7">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">n</font></span>)</span>
 ; <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">then </span><span class="kw">reconsider </span><font color="Maroon" title="c7">P1</font> = <span class="p1"> { <span class="default"> <font color="Olive" title="b1">q</font> where <font color="Olive" title="b1">q</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">n</font></span>)</span> :  for <font color="Olive" title="b2">r2</font> being   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a>  st <font color="Olive" title="b1">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="jordan2b.html#K1" title="JORDAN2B:func.1">|[</a><span class="default"><font color="Olive" title="b2">r2</font></span><a href="jordan2b.html#K1" title="JORDAN2B:func.1">]|</a></span> holds <br/><font color="Olive" title="b2">r2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <font color="Maroon" title="c3">r</font> </span> } </span>  as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">n</font></span>)</span> ;<br/><a NAME="E6:114_1_1_2_1_2"/>
<font color="Maroon" title="c1">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:114_1_1_2_1_2"><span class="lab"><font color="Green" title="E90">A17</font></span></a></span>;<br/><a NAME="E7:114_1_1_2_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c1">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T13" onmouseover="rs('nat_1/T13')" onmouseout="rh()">NAT_1:13</a></span>;<br/><a NAME="E8:114_1_1_2_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E91">A18</font></span>: 
<font color="Maroon" title="c1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:114_1_1"><span class="lab"><font color="Green" title="E87">A2</font></span></a>, <a class="ref" href="xxreal_0.html#T1" onmouseover="rs('xxreal_0/T1')" onmouseout="rh()">XXREAL_0:1</a></span>;<br/><a NAME="E9:114_1_1_2_1_2"/><span class="lab"><font color="Green" title="E92">A19</font></span>: 
<font color="Maroon" title="c5">P</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c7">P1</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c6">P2</font>
 <div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="114_1_1_2_1_2_3"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c8">z</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="c8">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">P</font> or <font color="Maroon" title="c8">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">P1</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c6">P2</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:114_1_1_2_1_2_3"/><span class="lab"><font color="Green" title="E93">A20</font></span>: 
<font color="Maroon" title="c8">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">P</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="c8">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">P1</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c6">P2</font></span><br/>

<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c9">q0</font> = <font color="Maroon" title="c8">z</font> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">n</font></span>)</span> ;<br/>
<span class="kw">consider </span><font color="Maroon" title="c10">r3</font> being   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a><span class="kw"> such that </span><br/><a NAME="E4:114_1_1_2_1_2_3"/><span class="lab"><font color="Green" title="E94">A21</font></span>: 
<font color="Maroon" title="c9">q0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><font color="Maroon" title="c10">r3</font></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:114_1_1_2_1_2"><span class="lab"><font color="Green" title="E91">A18</font></span></a>, <a class="ref" href="jordan2b.html#T20" onmouseover="rs('jordan2b/T20')" onmouseout="rh()">JORDAN2B:20</a></span>;<br/>
<a NAME="E5:114_1_1_2_1_2_3"/>
 not <font color="Maroon" title="c8">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">q</font> where <font color="Olive" title="b1">q</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">n</font></span>)</span> : <span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><font color="Olive" title="b1">q</font></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">r</font> </span> } </span> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:114_1_1_2_1_2_3"><span class="lab"><font color="Green" title="E93">A20</font></span></a>, <a class="ref" href="xboole_0.html#D5" onmouseover="rs('xboole_0/D5')" onmouseout="rh()">XBOOLE_0:def 5</a></span>;<br/>
<a NAME="E6:114_1_1_2_1_2_3"/><span class="kw">then </span>
<span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><font color="Maroon" title="c9">q0</font></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c3">r</font>
 
;<br/>
<a NAME="E7:114_1_1_2_1_2_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E95">A22</font></span>: 
<span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><font color="Maroon" title="c10">r3</font></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c3">r</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:114_1_1_2_1_2_3"><span class="lab"><font color="Green" title="E94">A21</font></span></a>, <a class="ref" href="jordan2c.html#T10" target="_self" onmouseover="rs('jordan2c/T10')" onmouseout="rh()">Th4</a></span>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="114_1_1_2_1_2_3_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="E1:114_1_1_2_1_2_3_1"/>
(  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <font color="Maroon" title="c3">r</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c10">r3</font> or <font color="Maroon" title="c10">r3</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c3">r</font> )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:114_1_1_2_1_2_3"><span class="lab"><font color="Green" title="E95">A22</font></span></a>, <a class="ref" href="seq_2.html#T1" onmouseover="rs('seq_2/T1')" onmouseout="rh()">SEQ_2:1</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="114_1_1_2_1_2_3_1_1"><span class="kw">suppose </span></a><a NAME="E1:114_1_1_2_1_2_3_1_1"/>
 <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <font color="Maroon" title="c3">r</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c10">r3</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="c8">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">P1</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c6">P2</font></span><br/><div class="add"><a NAME="E2:114_1_1_2_1_2_3_1_1"/><span class="kw">then </span>
 for <font color="Olive" title="b1">r2</font> being   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a>  st <font color="Maroon" title="c9">q0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="jordan2b.html#K1" title="JORDAN2B:func.1">|[</a><span class="default"><font color="Olive" title="b1">r2</font></span><a href="jordan2b.html#K1" title="JORDAN2B:func.1">]|</a></span> holds <br/><font color="Olive" title="b1">r2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <font color="Maroon" title="c3">r</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:114_1_1_2_1_2_3"><span class="lab"><font color="Green" title="E94">A21</font></span></a>, <a class="ref" href="jordan2b.html#T23" onmouseover="rs('jordan2b/T23')" onmouseout="rh()">JORDAN2B:23</a></span>;<br/><a NAME="E3:114_1_1_2_1_2_3_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c9">q0</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c7">P1</font>
 ;<br/><span class="kw">hence </span><a NAME="E4:114_1_1_2_1_2_3_1_1"/>
<font color="Maroon" title="c8">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">P1</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c6">P2</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D3" onmouseover="rs('xboole_0/D3')" onmouseout="rh()">XBOOLE_0:def 3</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="114_1_1_2_1_2_3_1_2"><span class="kw">suppose </span></a><a NAME="E1:114_1_1_2_1_2_3_1_2"/>
<font color="Maroon" title="c10">r3</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c3">r</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="c8">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">P1</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c6">P2</font></span><br/><div class="add"><a NAME="E2:114_1_1_2_1_2_3_1_2"/><span class="kw">then </span>
 for <font color="Olive" title="b1">r2</font> being   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a>  st <font color="Maroon" title="c9">q0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="jordan2b.html#K1" title="JORDAN2B:func.1">|[</a><span class="default"><font color="Olive" title="b1">r2</font></span><a href="jordan2b.html#K1" title="JORDAN2B:func.1">]|</a></span> holds <br/><font color="Olive" title="b1">r2</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c3">r</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:114_1_1_2_1_2_3"><span class="lab"><font color="Green" title="E94">A21</font></span></a>, <a class="ref" href="jordan2b.html#T23" onmouseover="rs('jordan2b/T23')" onmouseout="rh()">JORDAN2B:23</a></span>;<br/><a NAME="E3:114_1_1_2_1_2_3_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c9">q0</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c6">P2</font>
 ;<br/><span class="kw">hence </span><a NAME="E4:114_1_1_2_1_2_3_1_2"/>
<font color="Maroon" title="c8">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">P1</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c6">P2</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D3" onmouseover="rs('xboole_0/D3')" onmouseout="rh()">XBOOLE_0:def 3</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="E10:114_1_1_2_1_2"/>
<font color="Maroon" title="c7">P1</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c6">P2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">P</font>
 <div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="114_1_1_2_1_2_4"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c8">z</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="c8">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">P1</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c6">P2</font> or <font color="Maroon" title="c8">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">P</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:114_1_1_2_1_2_4"/><span class="lab"><font color="Green" title="E93">A23</font></span>: 
<font color="Maroon" title="c8">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">P1</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c6">P2</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="c8">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">P</font></span><br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="114_1_1_2_1_2_4_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="E1:114_1_1_2_1_2_4_1"/>
( <font color="Maroon" title="c8">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">P1</font> or <font color="Maroon" title="c8">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">P2</font> )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:114_1_1_2_1_2_4"><span class="lab"><font color="Green" title="E93">A23</font></span></a>, <a class="ref" href="xboole_0.html#D3" onmouseover="rs('xboole_0/D3')" onmouseout="rh()">XBOOLE_0:def 3</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="114_1_1_2_1_2_4_1_1"><span class="kw">suppose </span></a><a NAME="E1:114_1_1_2_1_2_4_1_1"/><span class="lab"><font color="Green" title="E94">A24</font></span>: 
<font color="Maroon" title="c8">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">P1</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="c8">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">P</font></span><br/><div class="add"><a NAME="E2:114_1_1_2_1_2_4_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E95">A25</font></span>: 
 ex <font color="Olive" title="b1">q</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">n</font></span>)</span> st <br/>( <font color="Olive" title="b1">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">z</font> &amp; (  for <font color="Olive" title="b2">r2</font> being   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a>  st <font color="Olive" title="b1">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="jordan2b.html#K1" title="JORDAN2B:func.1">|[</a><span class="default"><font color="Olive" title="b2">r2</font></span><a href="jordan2b.html#K1" title="JORDAN2B:func.1">]|</a></span> holds <br/><font color="Olive" title="b2">r2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <font color="Maroon" title="c3">r</font> ) )
 ;<br/><a NAME="E3:114_1_1_2_1_2_4_1_1"/>
 for <font color="Olive" title="b1">q2</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">n</font></span>)</span>  st <font color="Olive" title="b1">q2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">z</font> holds <br/><span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><font color="Olive" title="b1">q2</font></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c3">r</font>
 <div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="114_1_1_2_1_2_4_1_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c9">q2</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">n</font></span>)</span>; <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="c9">q2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">z</font> implies <span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><font color="Maroon" title="c9">q2</font></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c3">r</font> )</span><br/>

<span class="kw">consider </span><font color="Maroon" title="c10">r3</font> being   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a><span class="kw"> such that </span><br/><a NAME="E2:114_1_1_2_1_2_4_1_1_1"/><span class="lab"><font color="Green" title="E96">A26</font></span>: 
<font color="Maroon" title="c9">q2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><font color="Maroon" title="c10">r3</font></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:114_1_1_2_1_2"><span class="lab"><font color="Green" title="E91">A18</font></span></a>, <a class="ref" href="jordan2b.html#T20" onmouseover="rs('jordan2b/T20')" onmouseout="rh()">JORDAN2B:20</a></span>;<br/>
<span class="kw">assume </span><a NAME="E3:114_1_1_2_1_2_4_1_1_1"/><span class="lab"><font color="Green" title="E97">A27</font></span>: 
<font color="Maroon" title="c9">q2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">z</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"> <span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><font color="Maroon" title="c9">q2</font></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c3">r</font></span><br/>

<a NAME="E4:114_1_1_2_1_2_4_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E98">A28</font></span>: 
<font color="Maroon" title="c10">r3</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <font color="Maroon" title="c3">r</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:114_1_1_2_1_2_4_1_1"><span class="lab"><font color="Green" title="E95">A25</font></span></a>, <a class="txt" href="#E2:114_1_1_2_1_2_4_1_1_1"><span class="lab"><font color="Green" title="E96">A26</font></span></a></span>;<br/>
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="114_1_1_2_1_2_4_1_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"> ( ( <font color="Maroon" title="c3">r</font> <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; <span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><font color="Maroon" title="c10">r3</font></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c3">r</font> ) or ( <font color="Maroon" title="c3">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>  &amp; <span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><font color="Maroon" title="c10">r3</font></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c3">r</font> ) )</span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="114_1_1_2_1_2_4_1_1_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="E1:114_1_1_2_1_2_4_1_1_1_1_1"/>
( <font color="Maroon" title="c3">r</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  or <font color="Maroon" title="c3">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>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="114_1_1_2_1_2_4_1_1_1_1_1_1"><span class="kw">case </span></a><a NAME="E1:114_1_1_2_1_2_4_1_1_1_1_1_1"/>
<font color="Maroon" title="c3">r</font> <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"> <span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><font color="Maroon" title="c10">r3</font></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c3">r</font></span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:114_1_1_2_1_2_4_1_1_1_1_1_1"/>
<span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><font color="Maroon" title="c10">r3</font></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c3">r</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="complex1.html#T46" onmouseover="rs('complex1/T46')" onmouseout="rh()">COMPLEX1:46</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="114_1_1_2_1_2_4_1_1_1_1_1_2"><span class="kw">case </span></a><a NAME="E1:114_1_1_2_1_2_4_1_1_1_1_1_2"/>
<font color="Maroon" title="c3">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"> <span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><font color="Maroon" title="c10">r3</font></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c3">r</font></span><br/><div class="add"><a NAME="E2:114_1_1_2_1_2_4_1_1_1_1_1_2"/><span class="kw">then </span>
 <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <font color="Maroon" title="c3">r</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>
 ;<br/><a NAME="E3:114_1_1_2_1_2_4_1_1_1_1_1_2"/><span class="kw">then </span>
<span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><font color="Maroon" title="c10">r3</font></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <font color="Maroon" title="c10">r3</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:114_1_1_2_1_2_4_1_1_1"><span class="lab"><font color="Green" title="E98">A28</font></span></a>, <a class="ref" href="absvalue.html#T30" onmouseover="rs('absvalue/T30')" onmouseout="rh()">ABSVALUE:30</a></span>;<br/><span class="kw">hence </span><a NAME="E4:114_1_1_2_1_2_4_1_1_1_1_1_2"/>
<span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><font color="Maroon" title="c10">r3</font></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c3">r</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:114_1_1_2_1_2_4_1_1"><span class="lab"><font color="Green" title="E95">A25</font></span></a>, <a class="txt" href="#E3:114_1_1_2_1_2_4_1_1_1"><span class="lab"><font color="Green" title="E97">A27</font></span></a>, <a class="txt" href="#E2:114_1_1_2_1_2_4_1_1_1"><span class="lab"><font color="Green" title="E96">A26</font></span></a>, <a class="ref" href="xreal_1.html#T25" onmouseover="rs('xreal_1/T25')" onmouseout="rh()">XREAL_1:25</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>
<span class="kw">hence </span><a NAME="E6:114_1_1_2_1_2_4_1_1_1"/>
<span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><font color="Maroon" title="c9">q2</font></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c3">r</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:114_1_1_2_1_2_4_1_1_1"><span class="lab"><font color="Green" title="E96">A26</font></span></a>, <a class="ref" href="jordan2c.html#T10" target="_self" onmouseover="rs('jordan2c/T10')" onmouseout="rh()">Th4</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="E4:114_1_1_2_1_2_4_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E96">A29</font></span>: 
 not <font color="Maroon" title="c8">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">q2</font> where <font color="Olive" title="b1">q2</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">n</font></span>)</span> : <span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><font color="Olive" title="b1">q2</font></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">r</font> </span> } </span> 
 ;<br/><a NAME="E5:114_1_1_2_1_2_4_1_1"/>
<font color="Maroon" title="c8">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">n</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:114_1_1_2_1_2_4_1_1"><span class="lab"><font color="Green" title="E94">A24</font></span></a></span>;<br/><a NAME="E6:114_1_1_2_1_2_4_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c8">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="euclid.html#K1" title="EUCLID:func.1">REAL</a> <font color="Maroon" title="c1">n</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="euclid.html#T22" onmouseover="rs('euclid/T22')" onmouseout="rh()">EUCLID:22</a></span>;<br/><span class="kw">hence </span><a NAME="E7:114_1_1_2_1_2_4_1_1"/>
<font color="Maroon" title="c8">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">P</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:114_1_1_2_1_2_4_1_1"><span class="lab"><font color="Green" title="E96">A29</font></span></a>, <a class="ref" href="xboole_0.html#D5" onmouseover="rs('xboole_0/D5')" onmouseout="rh()">XBOOLE_0:def 5</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="114_1_1_2_1_2_4_1_2"><span class="kw">suppose </span></a><a NAME="E1:114_1_1_2_1_2_4_1_2"/><span class="lab"><font color="Green" title="E94">A30</font></span>: 
<font color="Maroon" title="c8">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">P2</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="c8">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">P</font></span><br/><div class="add"><a NAME="E2:114_1_1_2_1_2_4_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E95">A31</font></span>: 
 ex <font color="Olive" title="b1">q</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">n</font></span>)</span> st <br/>( <font color="Olive" title="b1">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">z</font> &amp; (  for <font color="Olive" title="b2">r2</font> being   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a>  st <font color="Olive" title="b1">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="jordan2b.html#K1" title="JORDAN2B:func.1">|[</a><span class="default"><font color="Olive" title="b2">r2</font></span><a href="jordan2b.html#K1" title="JORDAN2B:func.1">]|</a></span> holds <br/><font color="Olive" title="b2">r2</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c3">r</font> ) )
 ;<br/><a NAME="E3:114_1_1_2_1_2_4_1_2"/>
 for <font color="Olive" title="b1">q2</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">n</font></span>)</span>  st <font color="Olive" title="b1">q2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">z</font> holds <br/><span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><font color="Olive" title="b1">q2</font></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c3">r</font>
 <div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="114_1_1_2_1_2_4_1_2_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c9">q2</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">n</font></span>)</span>; <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="c9">q2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">z</font> implies <span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><font color="Maroon" title="c9">q2</font></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c3">r</font> )</span><br/>

<span class="kw">consider </span><font color="Maroon" title="c10">r3</font> being   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a><span class="kw"> such that </span><br/><a NAME="E2:114_1_1_2_1_2_4_1_2_1"/><span class="lab"><font color="Green" title="E96">A32</font></span>: 
<font color="Maroon" title="c9">q2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><font color="Maroon" title="c10">r3</font></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:114_1_1_2_1_2"><span class="lab"><font color="Green" title="E91">A18</font></span></a>, <a class="ref" href="jordan2b.html#T20" onmouseover="rs('jordan2b/T20')" onmouseout="rh()">JORDAN2B:20</a></span>;<br/>
<span class="kw">assume </span><a NAME="E3:114_1_1_2_1_2_4_1_2_1"/>
<font color="Maroon" title="c9">q2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">z</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"> <span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><font color="Maroon" title="c9">q2</font></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c3">r</font></span><br/>

<a NAME="E4:114_1_1_2_1_2_4_1_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E97">A33</font></span>: 
<font color="Maroon" title="c10">r3</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c3">r</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:114_1_1_2_1_2_4_1_2"><span class="lab"><font color="Green" title="E95">A31</font></span></a>, <a class="txt" href="#E2:114_1_1_2_1_2_4_1_2_1"><span class="lab"><font color="Green" title="E96">A32</font></span></a></span>;<br/>
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="114_1_1_2_1_2_4_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"> <span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><font color="Maroon" title="c10">r3</font></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c3">r</font></span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="114_1_1_2_1_2_4_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="E1:114_1_1_2_1_2_4_1_2_1_1_1"/>
( <font color="Maroon" title="c3">r</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  or <font color="Maroon" title="c3">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>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="114_1_1_2_1_2_4_1_2_1_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:114_1_1_2_1_2_4_1_2_1_1_1_1"/>
<font color="Maroon" title="c3">r</font> <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"> <span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><font color="Maroon" title="c10">r3</font></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c3">r</font></span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:114_1_1_2_1_2_4_1_2_1_1_1_1"/>
<span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><font color="Maroon" title="c10">r3</font></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c3">r</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="complex1.html#T46" onmouseover="rs('complex1/T46')" onmouseout="rh()">COMPLEX1:46</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="114_1_1_2_1_2_4_1_2_1_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:114_1_1_2_1_2_4_1_2_1_1_1_2"/>
<font color="Maroon" title="c3">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"> <span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><font color="Maroon" title="c10">r3</font></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c3">r</font></span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:114_1_1_2_1_2_4_1_2_1_1_1_2"/>
<span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><font color="Maroon" title="c10">r3</font></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c3">r</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:114_1_1_2_1_2_4_1_2_1"><span class="lab"><font color="Green" title="E97">A33</font></span></a>, <a class="ref" href="absvalue.html#D1" onmouseover="rs('absvalue/D1')" onmouseout="rh()">ABSVALUE:def 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>
<span class="kw">hence </span><a NAME="E6:114_1_1_2_1_2_4_1_2_1"/>
<span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><font color="Maroon" title="c9">q2</font></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c3">r</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:114_1_1_2_1_2_4_1_2_1"><span class="lab"><font color="Green" title="E96">A32</font></span></a>, <a class="ref" href="jordan2c.html#T10" target="_self" onmouseover="rs('jordan2c/T10')" onmouseout="rh()">Th4</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="E4:114_1_1_2_1_2_4_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E96">A34</font></span>: 
 not <font color="Maroon" title="c8">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">q2</font> where <font color="Olive" title="b1">q2</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">n</font></span>)</span> : <span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><font color="Olive" title="b1">q2</font></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">r</font> </span> } </span> 
 ;<br/><a NAME="E5:114_1_1_2_1_2_4_1_2"/>
<font color="Maroon" title="c8">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">n</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:114_1_1_2_1_2_4_1_2"><span class="lab"><font color="Green" title="E94">A30</font></span></a></span>;<br/><a NAME="E6:114_1_1_2_1_2_4_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c8">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="euclid.html#K1" title="EUCLID:func.1">REAL</a> <font color="Maroon" title="c1">n</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="euclid.html#T22" onmouseover="rs('euclid/T22')" onmouseout="rh()">EUCLID:22</a></span>;<br/><span class="kw">hence </span><a NAME="E7:114_1_1_2_1_2_4_1_2"/>
<font color="Maroon" title="c8">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">P</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:114_1_1_2_1_2_4_1_2"><span class="lab"><font color="Green" title="E96">A34</font></span></a>, <a class="ref" href="xboole_0.html#D5" onmouseover="rs('xboole_0/D5')" onmouseout="rh()">XBOOLE_0:def 5</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="E11:114_1_1_2_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E93">A35</font></span>: 
<font color="Maroon" title="c5">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">P1</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c6">P2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:114_1_1_2_1_2"><span class="lab"><font color="Green" title="E92">A19</font></span></a></span>;<br/><a NAME="E12:114_1_1_2_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c6">P2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">P</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T7" onmouseover="rs('xboole_1/T7')" onmouseout="rh()">XBOOLE_1:7</a></span>;<br/><a NAME="E13:114_1_1_2_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E94">A36</font></span>: 
 <a href="connsp_3.html#K4" title="CONNSP_3:func.4">Down</a> (<font color="Maroon" title="c6">P2</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a></span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">P2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:114_1_1"><span class="lab"><font color="Green" title="E88">A3</font></span></a>, <a class="ref" href="xboole_1.html#T1" onmouseover="rs('xboole_1/T1')" onmouseout="rh()">XBOOLE_1:1</a>, <a class="ref" href="xboole_1.html#T28" onmouseover="rs('xboole_1/T28')" onmouseout="rh()">XBOOLE_1:28</a></span>;<br/><a NAME="E14:114_1_1_2_1_2"/>
 for <font color="Olive" title="b1">w1</font>, <font color="Olive" title="b2">w2</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">n</font></span>)</span>  st <font color="Olive" title="b1">w1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c6">P2</font> &amp; <font color="Olive" title="b2">w2</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c6">P2</font> holds <br/> <a href="rltopsp1.html#K1" title="RLTOPSP1:func.1">LSeg</a> (<font color="Olive" title="b1">w1</font>,<font color="Olive" title="b2">w2</font>) <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c6">P2</font>
 <div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="114_1_1_2_1_2_5"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c8">w1</font>, <font color="Maroon" title="c9">w2</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">n</font></span>)</span>; <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="c8">w1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c6">P2</font> &amp; <font color="Maroon" title="c9">w2</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c6">P2</font> implies  <a href="rltopsp1.html#K1" title="RLTOPSP1:func.1">LSeg</a> (<font color="Maroon" title="c8">w1</font>,<font color="Maroon" title="c9">w2</font>) <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c6">P2</font> )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:114_1_1_2_1_2_5"/><span class="lab"><font color="Green" title="E95">A37</font></span>: 
<font color="Maroon" title="c8">w1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c6">P2</font>
 <span class="kw">and </span><br/><a NAME="E2:114_1_1_2_1_2_5"/><span class="lab"><font color="Green" title="E96">A38</font></span>: 
<font color="Maroon" title="c9">w2</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c6">P2</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">  <a href="rltopsp1.html#K1" title="RLTOPSP1:func.1">LSeg</a> (<font color="Maroon" title="c8">w1</font>,<font color="Maroon" title="c9">w2</font>) <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c6">P2</font></span><br/>

<a NAME="E3:114_1_1_2_1_2_5"/><span class="lab"><font color="Green" title="E97">A39</font></span>: 
 ex <font color="Olive" title="b1">q2</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">n</font></span>)</span> st <br/>( <font color="Olive" title="b1">q2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">w2</font> &amp; (  for <font color="Olive" title="b2">r2</font> being   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a>  st <font color="Olive" title="b1">q2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="jordan2b.html#K1" title="JORDAN2B:func.1">|[</a><span class="default"><font color="Olive" title="b2">r2</font></span><a href="jordan2b.html#K1" title="JORDAN2B:func.1">]|</a></span> holds <br/><font color="Olive" title="b2">r2</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c3">r</font> ) )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:114_1_1_2_1_2_5"><span class="lab"><font color="Green" title="E96">A38</font></span></a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c10">r3</font> being   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a><span class="kw"> such that </span><br/><a NAME="E5:114_1_1_2_1_2_5"/><span class="lab"><font color="Green" title="E98">A40</font></span>: 
<font color="Maroon" title="c8">w1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><font color="Maroon" title="c10">r3</font></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:114_1_1_2_1_2"><span class="lab"><font color="Green" title="E91">A18</font></span></a>, <a class="ref" href="jordan2b.html#T20" onmouseover="rs('jordan2b/T20')" onmouseout="rh()">JORDAN2B:20</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c11">r4</font> being   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a><span class="kw"> such that </span><br/><a NAME="E7:114_1_1_2_1_2_5"/><span class="lab"><font color="Green" title="E99">A41</font></span>: 
<font color="Maroon" title="c9">w2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><font color="Maroon" title="c11">r4</font></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:114_1_1_2_1_2"><span class="lab"><font color="Green" title="E91">A18</font></span></a>, <a class="ref" href="jordan2b.html#T20" onmouseover="rs('jordan2b/T20')" onmouseout="rh()">JORDAN2B:20</a></span>;<br/>
<a NAME="E8:114_1_1_2_1_2_5"/><span class="lab"><font color="Green" title="E100">A42</font></span>: 
 ex <font color="Olive" title="b1">q1</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">n</font></span>)</span> st <br/>( <font color="Olive" title="b1">q1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">w1</font> &amp; (  for <font color="Olive" title="b2">r2</font> being   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a>  st <font color="Olive" title="b1">q1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="jordan2b.html#K1" title="JORDAN2B:func.1">|[</a><span class="default"><font color="Olive" title="b2">r2</font></span><a href="jordan2b.html#K1" title="JORDAN2B:func.1">]|</a></span> holds <br/><font color="Olive" title="b2">r2</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c3">r</font> ) )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:114_1_1_2_1_2_5"><span class="lab"><font color="Green" title="E95">A37</font></span></a></span>;<br/>
<span class="kw">thus </span><a NAME="E9:114_1_1_2_1_2_5"/>
 <a href="rltopsp1.html#K1" title="RLTOPSP1:func.1">LSeg</a> (<font color="Maroon" title="c8">w1</font>,<font color="Maroon" title="c9">w2</font>) <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c6">P2</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><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="114_1_1_2_1_2_5_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c12">z</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="c12">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rltopsp1.html#K1" title="RLTOPSP1:func.1">LSeg</a> (<font color="Maroon" title="c8">w1</font>,<font color="Maroon" title="c9">w2</font>) or <font color="Maroon" title="c12">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">P2</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:114_1_1_2_1_2_5_1"/>
<font color="Maroon" title="c12">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rltopsp1.html#K1" title="RLTOPSP1:func.1">LSeg</a> (<font color="Maroon" title="c8">w1</font>,<font color="Maroon" title="c9">w2</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="c12">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">P2</font></span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c13">r2</font> being   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a><span class="kw"> such that </span><br/><a NAME="E3:114_1_1_2_1_2_5_1"/><span class="lab"><font color="Green" title="E101">A43</font></span>: 
<font color="Maroon" title="c12">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c13">r2</font></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c8">w1</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">r2</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c9">w2</font></span>)</span>
 <span class="kw">and </span><br/><a NAME="E4:114_1_1_2_1_2_5_1"/><span class="lab"><font color="Green" title="E102">A44</font></span>: 
 <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="c13">r2</font>
 <span class="kw">and </span><br/><a NAME="E5:114_1_1_2_1_2_5_1"/><span class="lab"><font color="Green" title="E103">A45</font></span>: 
<font color="Maroon" title="c13">r2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1
 ;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c14">q4</font> = <font color="Maroon" title="c12">z</font> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">n</font></span>)</span> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:114_1_1_2_1_2_5_1"><span class="lab"><font color="Green" title="E101">A43</font></span></a></span>;<br/>
<a NAME="E7:114_1_1_2_1_2_5_1"/>
( <span class="p1">(<span class="default">1 <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c13">r2</font></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c8">w1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="jordan2b.html#K1" title="JORDAN2B:func.1">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c13">r2</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c10">r3</font></span>)</span></span><a href="jordan2b.html#K1" title="JORDAN2B:func.1">]|</a></span> &amp; <font color="Maroon" title="c13">r2</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c9">w2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="jordan2b.html#K1" title="JORDAN2B:func.1">|[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c13">r2</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c11">r4</font></span>)</span></span><a href="jordan2b.html#K1" title="JORDAN2B:func.1">]|</a></span> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:114_1_1_2_1_2"><span class="lab"><font color="Green" title="E91">A18</font></span></a>, <a class="txt" href="#E5:114_1_1_2_1_2_5"><span class="lab"><font color="Green" title="E98">A40</font></span></a>, <a class="txt" href="#E7:114_1_1_2_1_2_5"><span class="lab"><font color="Green" title="E99">A41</font></span></a>, <a class="ref" href="jordan2b.html#T21" onmouseover="rs('jordan2b/T21')" onmouseout="rh()">JORDAN2B:21</a></span>;<br/>
<a NAME="E8:114_1_1_2_1_2_5_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E104">A46</font></span>: 
<font color="Maroon" title="c12">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="jordan2b.html#K1" title="JORDAN2B:func.1">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default">1 <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c13">r2</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c10">r3</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c13">r2</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c11">r4</font></span>)</span></span>)</span></span><a href="jordan2b.html#K1" title="JORDAN2B:func.1">]|</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:114_1_1_2_1_2"><span class="lab"><font color="Green" title="E91">A18</font></span></a>, <a class="txt" href="#E3:114_1_1_2_1_2_5_1"><span class="lab"><font color="Green" title="E101">A43</font></span></a>, <a class="ref" href="jordan2b.html#T22" onmouseover="rs('jordan2b/T22')" onmouseout="rh()">JORDAN2B:22</a></span>;<br/>
<a NAME="E9:114_1_1_2_1_2_5_1"/>
 for <font color="Olive" title="b1">r5</font> being   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a>  st <font color="Maroon" title="c14">q4</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="jordan2b.html#K1" title="JORDAN2B:func.1">|[</a><span class="default"><font color="Olive" title="b1">r5</font></span><a href="jordan2b.html#K1" title="JORDAN2B:func.1">]|</a></span> holds <br/><font color="Olive" title="b1">r5</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c3">r</font>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="114_1_1_2_1_2_5_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c15">r5</font> be   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</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="c14">q4</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="jordan2b.html#K1" title="JORDAN2B:func.1">|[</a><span class="default"><font color="Maroon" title="c15">r5</font></span><a href="jordan2b.html#K1" title="JORDAN2B:func.1">]|</a></span> implies <font color="Maroon" title="c15">r5</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c3">r</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:114_1_1_2_1_2_5_1_1"/>
<font color="Maroon" title="c14">q4</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="jordan2b.html#K1" title="JORDAN2B:func.1">|[</a><span class="default"><font color="Maroon" title="c15">r5</font></span><a href="jordan2b.html#K1" title="JORDAN2B:func.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"> <font color="Maroon" title="c15">r5</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c3">r</font></span><br/>

<a NAME="E2:114_1_1_2_1_2_5_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E105">A47</font></span>: 
<font color="Maroon" title="c15">r5</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c13">r2</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c10">r3</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">r2</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c11">r4</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:114_1_1_2_1_2_5_1"><span class="lab"><font color="Green" title="E104">A46</font></span></a>, <a class="ref" href="jordan2b.html#T23" onmouseover="rs('jordan2b/T23')" onmouseout="rh()">JORDAN2B:23</a></span>;<br/>
<a NAME="E3:114_1_1_2_1_2_5_1_1"/>
1 <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c13">r2</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="#E5:114_1_1_2_1_2_5_1"><span class="lab"><font color="Green" title="E103">A45</font></span></a>, <a class="ref" href="xreal_1.html#T48" onmouseover="rs('xreal_1/T48')" onmouseout="rh()">XREAL_1:48</a></span>;<br/>
<a NAME="E4:114_1_1_2_1_2_5_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E106">A48</font></span>: 
<span class="p1">(<span class="default">1 <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c13">r2</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c10">r3</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <span class="p1">(<span class="default">1 <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c13">r2</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c3">r</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:114_1_1_2_1_2_5"><span class="lab"><font color="Green" title="E100">A42</font></span></a>, <a class="txt" href="#E5:114_1_1_2_1_2_5"><span class="lab"><font color="Green" title="E98">A40</font></span></a>, <a class="ref" href="xreal_1.html#T64" onmouseover="rs('xreal_1/T64')" onmouseout="rh()">XREAL_1:64</a></span>;<br/>
<a NAME="E5:114_1_1_2_1_2_5_1_1"/>
( <font color="Maroon" title="c13">r2</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c11">r4</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c13">r2</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c3">r</font> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c13">r2</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c3">r</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">r2</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c3">r</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">r</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:114_1_1_2_1_2_5"><span class="lab"><font color="Green" title="E97">A39</font></span></a>, <a class="txt" href="#E7:114_1_1_2_1_2_5"><span class="lab"><font color="Green" title="E99">A41</font></span></a>, <a class="txt" href="#E4:114_1_1_2_1_2_5_1"><span class="lab"><font color="Green" title="E102">A44</font></span></a>, <a class="ref" href="xreal_1.html#T64" onmouseover="rs('xreal_1/T64')" onmouseout="rh()">XREAL_1:64</a></span>;<br/>
<span class="kw">hence </span><a NAME="E6:114_1_1_2_1_2_5_1_1"/>
<font color="Maroon" title="c15">r5</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c3">r</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:114_1_1_2_1_2_5_1_1"><span class="lab"><font color="Green" title="E105">A47</font></span></a>, <a class="txt" href="#E4:114_1_1_2_1_2_5_1_1"><span class="lab"><font color="Green" title="E106">A48</font></span></a>, <a class="ref" href="xreal_1.html#T7" onmouseover="rs('xreal_1/T7')" onmouseout="rh()">XREAL_1:7</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="E10:114_1_1_2_1_2_5_1"/>
<font color="Maroon" title="c12">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">P2</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>


</div><span class="kw">end;</span></div><a NAME="E15:114_1_1_2_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c6">P2</font> is  <a href="convex1.html#V1" title="CONVEX1:attr.1">convex</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="jordan1.html#D1" onmouseover="rs('jordan1/D1')" onmouseout="rh()">JORDAN1:def 1</a></span>;<br/><a NAME="E16:114_1_1_2_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E95">A49</font></span>: 
 <a href="connsp_3.html#K4" title="CONNSP_3:func.4">Down</a> (<font color="Maroon" title="c6">P2</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a></span>)</span>) is  <a href="connsp_1.html#V2" title="CONNSP_1:attr.2">connected</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:114_1_1_2_1_2"><span class="lab"><font color="Green" title="E94">A36</font></span></a>, <a class="ref" href="connsp_1.html#T46" onmouseover="rs('connsp_1/T46')" onmouseout="rh()">CONNSP_1:46</a></span>;<br/><a NAME="E17:114_1_1_2_1_2"/>
<font color="Maroon" title="c7">P1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">P</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:114_1_1_2_1_2"><span class="lab"><font color="Green" title="E93">A35</font></span></a>, <a class="ref" href="xboole_1.html#T7" onmouseover="rs('xboole_1/T7')" onmouseout="rh()">XBOOLE_1:7</a></span>;<br/><a NAME="E18:114_1_1_2_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E96">A50</font></span>: 
 <a href="connsp_3.html#K4" title="CONNSP_3:func.4">Down</a> (<font color="Maroon" title="c7">P1</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a></span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">P1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:114_1_1"><span class="lab"><font color="Green" title="E88">A3</font></span></a>, <a class="ref" href="xboole_1.html#T1" onmouseover="rs('xboole_1/T1')" onmouseout="rh()">XBOOLE_1:1</a>, <a class="ref" href="xboole_1.html#T28" onmouseover="rs('xboole_1/T28')" onmouseout="rh()">XBOOLE_1:28</a></span>;<br/><div><span class="lab"><font color="Green" title="E97">A51</font></span>: <a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="114_1_1_2_1_2_6"><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="c6">P2</font> is  <a href="rltopsp1.html#V9" title="RLTOPSP1:attr.9">bounded</a> </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:114_1_1_2_1_2_6"/>
<font color="Maroon" title="c6">P2</font> is  <a href="rltopsp1.html#V9" title="RLTOPSP1:attr.9">bounded</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/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c8">r</font> being   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a><span class="kw"> such that </span><br/><a NAME="E3:114_1_1_2_1_2_6"/><span class="lab"><font color="Green" title="E98">A52</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 <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">n</font></span>)</span>  st <font color="Olive" title="b1">q</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c6">P2</font> holds <br/><span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><font color="Olive" title="b1">q</font></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c8">r</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="jordan2c.html#T34" target="_self" onmouseover="rs('jordan2c/T34')" onmouseout="rh()">Th21</a></span>;<br/><a NAME="E4:114_1_1_2_1_2_6"/>
(  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><font color="Maroon" title="c8">r</font></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span> &amp;  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><font color="Maroon" title="c3">r</font></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="complex1.html#T46" onmouseover="rs('complex1/T46')" onmouseout="rh()">COMPLEX1:46</a></span>;<br/><a NAME="E5:114_1_1_2_1_2_6"/><span class="kw">then </span><span class="lab"><font color="Green" title="E99">A53</font></span>: 
<span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><font color="Maroon" title="c8">r</font></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p3"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><font color="Maroon" title="c3">r</font></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span></span>)</span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><font color="Maroon" title="c8">r</font></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><font color="Maroon" title="c3">r</font></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="absvalue.html#D1" onmouseover="rs('absvalue/D1')" onmouseout="rh()">ABSVALUE:def 1</a></span>;<br/><span class="kw">set </span><font color="Maroon" title="c9">p3</font> = <span class="p1"><a href="jordan2b.html#K1" title="JORDAN2B:func.1">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><font color="Maroon" title="c8">r</font></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p3"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><font color="Maroon" title="c3">r</font></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span></span>)</span></span><a href="jordan2b.html#K1" title="JORDAN2B:func.1">]|</a></span>;<br/><a NAME="E6:114_1_1_2_1_2_6"/><span class="lab"><font color="Green" title="E100">A54</font></span>: 
<span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><font color="Maroon" title="c8">r</font></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><font color="Maroon" title="c8">r</font></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><font color="Maroon" title="c3">r</font></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="complex1.html#T46" onmouseover="rs('complex1/T46')" onmouseout="rh()">COMPLEX1:46</a>, <a class="ref" href="xreal_1.html#T31" onmouseover="rs('xreal_1/T31')" onmouseout="rh()">XREAL_1:31</a></span>;<br/><a NAME="E7:114_1_1_2_1_2_6"/>
 for <font color="Olive" title="b1">r5</font> being   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a>  st <span class="p1"><a href="jordan2b.html#K1" title="JORDAN2B:func.1">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><font color="Maroon" title="c8">r</font></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p3"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><font color="Maroon" title="c3">r</font></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span></span>)</span></span><a href="jordan2b.html#K1" title="JORDAN2B:func.1">]|</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="jordan2b.html#K1" title="JORDAN2B:func.1">|[</a><span class="default"><font color="Olive" title="b1">r5</font></span><a href="jordan2b.html#K1" title="JORDAN2B:func.1">]|</a></span> holds <br/><font color="Olive" title="b1">r5</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c3">r</font>
 <div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="114_1_1_2_1_2_6_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c10">r5</font> be   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</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"> ( <span class="p1"><a href="jordan2b.html#K1" title="JORDAN2B:func.1">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><font color="Maroon" title="c8">r</font></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p3"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><font color="Maroon" title="c3">r</font></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span></span>)</span></span><a href="jordan2b.html#K1" title="JORDAN2B:func.1">]|</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="jordan2b.html#K1" title="JORDAN2B:func.1">|[</a><span class="default"><font color="Maroon" title="c10">r5</font></span><a href="jordan2b.html#K1" title="JORDAN2B:func.1">]|</a></span> implies <font color="Maroon" title="c10">r5</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c3">r</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:114_1_1_2_1_2_6_1"/>
<span class="p1"><a href="jordan2b.html#K1" title="JORDAN2B:func.1">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><font color="Maroon" title="c8">r</font></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p3"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><font color="Maroon" title="c3">r</font></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span></span>)</span></span><a href="jordan2b.html#K1" title="JORDAN2B:func.1">]|</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="jordan2b.html#K1" title="JORDAN2B:func.1">|[</a><span class="default"><font color="Maroon" title="c10">r5</font></span><a href="jordan2b.html#K1" title="JORDAN2B:func.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"> <font color="Maroon" title="c10">r5</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c3">r</font></span><br/>

<a NAME="E2:114_1_1_2_1_2_6_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E101">A55</font></span>: 
<font color="Maroon" title="c10">r5</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><font color="Maroon" title="c8">r</font></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><font color="Maroon" title="c3">r</font></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="jordan2b.html#T23" onmouseover="rs('jordan2b/T23')" onmouseout="rh()">JORDAN2B:23</a></span>;<br/>
<a NAME="E3:114_1_1_2_1_2_6_1"/>
( <font color="Maroon" title="c3">r</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><font color="Maroon" title="c3">r</font></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span> &amp; <span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><font color="Maroon" title="c3">r</font></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><font color="Maroon" title="c3">r</font></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><font color="Maroon" title="c8">r</font></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="absvalue.html#T4" onmouseover="rs('absvalue/T4')" onmouseout="rh()">ABSVALUE:4</a>, <a class="ref" href="complex1.html#T46" onmouseover="rs('complex1/T46')" onmouseout="rh()">COMPLEX1:46</a>, <a class="ref" href="xreal_1.html#T31" onmouseover="rs('xreal_1/T31')" onmouseout="rh()">XREAL_1:31</a></span>;<br/>
<span class="kw">hence </span><a NAME="E4:114_1_1_2_1_2_6_1"/>
<font color="Maroon" title="c10">r5</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c3">r</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:114_1_1_2_1_2_6_1"><span class="lab"><font color="Green" title="E101">A55</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><a NAME="E8:114_1_1_2_1_2_6"/><span class="kw">then </span><span class="lab"><font color="Green" title="E101">A56</font></span>: 
<span class="p1"><a href="jordan2b.html#K1" title="JORDAN2B:func.1">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><font color="Maroon" title="c8">r</font></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p3"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><font color="Maroon" title="c3">r</font></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span></span>)</span></span><a href="jordan2b.html#K1" title="JORDAN2B:func.1">]|</a></span> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c6">P2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:114_1_1_2_1_2"><span class="lab"><font color="Green" title="E91">A18</font></span></a></span>;<br/><a NAME="E9:114_1_1_2_1_2_6"/>
( <span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><span class="p2"><a href="jordan2b.html#K1" title="JORDAN2B:func.1">|[</a><span class="default"><span class="p3">(<span class="default"><span class="p4"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><font color="Maroon" title="c8">r</font></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p4"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><font color="Maroon" title="c3">r</font></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span></span>)</span></span><a href="jordan2b.html#K1" title="JORDAN2B:func.1">]|</a></span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><font color="Maroon" title="c8">r</font></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p3"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><font color="Maroon" title="c3">r</font></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span></span>)</span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span> &amp; <font color="Maroon" title="c8">r</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><font color="Maroon" title="c8">r</font></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="jordan2c.html#T10" target="_self" onmouseover="rs('jordan2c/T10')" onmouseout="rh()">Th4</a>, <a class="ref" href="absvalue.html#T4" onmouseover="rs('absvalue/T4')" onmouseout="rh()">ABSVALUE:4</a></span>;<br/><span class="kw">hence </span><a NAME="E10:114_1_1_2_1_2_6"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:114_1_1_2_1_2_6"><span class="lab"><font color="Green" title="E98">A52</font></span></a>, <a class="txt" href="#E8:114_1_1_2_1_2_6"><span class="lab"><font color="Green" title="E101">A56</font></span></a>, <a class="txt" href="#E5:114_1_1_2_1_2_6"><span class="lab"><font color="Green" title="E99">A53</font></span></a>, <a class="txt" href="#E6:114_1_1_2_1_2_6"><span class="lab"><font color="Green" title="E100">A54</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><span class="lab"><font color="Green" title="E98">A57</font></span>: <a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="114_1_1_2_1_2_7"><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="c7">P1</font> is  <a href="rltopsp1.html#V9" title="RLTOPSP1:attr.9">bounded</a> </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:114_1_1_2_1_2_7"/>
<font color="Maroon" title="c7">P1</font> is  <a href="rltopsp1.html#V9" title="RLTOPSP1:attr.9">bounded</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/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c8">r</font> being   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a><span class="kw"> such that </span><br/><a NAME="E3:114_1_1_2_1_2_7"/><span class="lab"><font color="Green" title="E99">A58</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 <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">n</font></span>)</span>  st <font color="Olive" title="b1">q</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c7">P1</font> holds <br/><span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><font color="Olive" title="b1">q</font></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c8">r</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="jordan2c.html#T34" target="_self" onmouseover="rs('jordan2c/T34')" onmouseout="rh()">Th21</a></span>;<br/><a NAME="E4:114_1_1_2_1_2_7"/>
(  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><font color="Maroon" title="c8">r</font></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span> &amp;  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><font color="Maroon" title="c3">r</font></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="complex1.html#T46" onmouseover="rs('complex1/T46')" onmouseout="rh()">COMPLEX1:46</a></span>;<br/><a NAME="E5:114_1_1_2_1_2_7"/><span class="kw">then </span><span class="lab"><font color="Green" title="E100">A59</font></span>: 
<span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><font color="Maroon" title="c8">r</font></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p3"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><font color="Maroon" title="c3">r</font></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span></span>)</span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><font color="Maroon" title="c8">r</font></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><font color="Maroon" title="c3">r</font></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="absvalue.html#D1" onmouseover="rs('absvalue/D1')" onmouseout="rh()">ABSVALUE:def 1</a></span>;<br/><span class="kw">set </span><font color="Maroon" title="c9">p3</font> = <span class="p1"><a href="jordan2b.html#K1" title="JORDAN2B:func.1">|[</a><span class="default"><span class="p2">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p3">(<span class="default"><span class="p4"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><font color="Maroon" title="c8">r</font></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p4"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><font color="Maroon" title="c3">r</font></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span></span>)</span></span>)</span></span><a href="jordan2b.html#K1" title="JORDAN2B:func.1">]|</a></span>;<br/><a NAME="E6:114_1_1_2_1_2_7"/><span class="lab"><font color="Green" title="E101">A60</font></span>: 
( <font color="Maroon" title="c8">r</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><font color="Maroon" title="c8">r</font></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span> &amp; <span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><font color="Maroon" title="c8">r</font></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><font color="Maroon" title="c8">r</font></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><font color="Maroon" title="c3">r</font></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="absvalue.html#T4" onmouseover="rs('absvalue/T4')" onmouseout="rh()">ABSVALUE:4</a>, <a class="ref" href="complex1.html#T46" onmouseover="rs('complex1/T46')" onmouseout="rh()">COMPLEX1:46</a>, <a class="ref" href="xreal_1.html#T31" onmouseover="rs('xreal_1/T31')" onmouseout="rh()">XREAL_1:31</a></span>;<br/><a NAME="E7:114_1_1_2_1_2_7"/>
 for <font color="Olive" title="b1">r5</font> being   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a>  st <span class="p1"><a href="jordan2b.html#K1" title="JORDAN2B:func.1">|[</a><span class="default"><span class="p2">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p3">(<span class="default"><span class="p4"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><font color="Maroon" title="c8">r</font></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p4"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><font color="Maroon" title="c3">r</font></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span></span>)</span></span>)</span></span><a href="jordan2b.html#K1" title="JORDAN2B:func.1">]|</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="jordan2b.html#K1" title="JORDAN2B:func.1">|[</a><span class="default"><font color="Olive" title="b1">r5</font></span><a href="jordan2b.html#K1" title="JORDAN2B:func.1">]|</a></span> holds <br/><font color="Olive" title="b1">r5</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <font color="Maroon" title="c3">r</font>
 <div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="114_1_1_2_1_2_7_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c10">r5</font> be   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</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"> ( <span class="p1"><a href="jordan2b.html#K1" title="JORDAN2B:func.1">|[</a><span class="default"><span class="p2">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p3">(<span class="default"><span class="p4"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><font color="Maroon" title="c8">r</font></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p4"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><font color="Maroon" title="c3">r</font></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span></span>)</span></span>)</span></span><a href="jordan2b.html#K1" title="JORDAN2B:func.1">]|</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="jordan2b.html#K1" title="JORDAN2B:func.1">|[</a><span class="default"><font color="Maroon" title="c10">r5</font></span><a href="jordan2b.html#K1" title="JORDAN2B:func.1">]|</a></span> implies <font color="Maroon" title="c10">r5</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <font color="Maroon" title="c3">r</font> )</span><br/>

<a NAME="E1:114_1_1_2_1_2_7_1"/>
<font color="Maroon" title="c3">r</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><font color="Maroon" title="c3">r</font></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="absvalue.html#T4" onmouseover="rs('absvalue/T4')" onmouseout="rh()">ABSVALUE:4</a></span>;<br/>
<a NAME="E2:114_1_1_2_1_2_7_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E102">A61</font></span>: 
 <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><font color="Maroon" title="c3">r</font></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <font color="Maroon" title="c3">r</font>
 
<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="E3:114_1_1_2_1_2_7_1"/>
<span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><font color="Maroon" title="c3">r</font></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><font color="Maroon" title="c3">r</font></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><font color="Maroon" title="c8">r</font></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="complex1.html#T46" onmouseover="rs('complex1/T46')" onmouseout="rh()">COMPLEX1:46</a>, <a class="ref" href="xreal_1.html#T31" onmouseover="rs('xreal_1/T31')" onmouseout="rh()">XREAL_1:31</a></span>;<br/>
<a NAME="E4:114_1_1_2_1_2_7_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E103">A62</font></span>: 
 <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><font color="Maroon" title="c3">r</font></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p1">(<span class="default"><span class="p2"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><font color="Maroon" title="c3">r</font></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p2"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><font color="Maroon" title="c8">r</font></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></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/>
<span class="kw">assume </span><a NAME="E5:114_1_1_2_1_2_7_1"/>
<span class="p1"><a href="jordan2b.html#K1" title="JORDAN2B:func.1">|[</a><span class="default"><span class="p2">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p3">(<span class="default"><span class="p4"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><font color="Maroon" title="c8">r</font></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p4"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><font color="Maroon" title="c3">r</font></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span></span>)</span></span>)</span></span><a href="jordan2b.html#K1" title="JORDAN2B:func.1">]|</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="jordan2b.html#K1" title="JORDAN2B:func.1">|[</a><span class="default"><font color="Maroon" title="c10">r5</font></span><a href="jordan2b.html#K1" title="JORDAN2B:func.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"> <font color="Maroon" title="c10">r5</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <font color="Maroon" title="c3">r</font></span><br/>

<a NAME="E6:114_1_1_2_1_2_7_1"/><span class="kw">then </span>
<font color="Maroon" title="c10">r5</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p1">(<span class="default"><span class="p2"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><font color="Maroon" title="c8">r</font></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p2"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><font color="Maroon" title="c3">r</font></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="jordan2b.html#T23" onmouseover="rs('jordan2b/T23')" onmouseout="rh()">JORDAN2B:23</a></span>;<br/>
<span class="kw">hence </span><a NAME="E7:114_1_1_2_1_2_7_1"/>
<font color="Maroon" title="c10">r5</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <font color="Maroon" title="c3">r</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:114_1_1_2_1_2_7_1"><span class="lab"><font color="Green" title="E102">A61</font></span></a>, <a class="txt" href="#E4:114_1_1_2_1_2_7_1"><span class="lab"><font color="Green" title="E103">A62</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><a NAME="E8:114_1_1_2_1_2_7"/><span class="kw">then </span><span class="lab"><font color="Green" title="E102">A63</font></span>: 
<span class="p1"><a href="jordan2b.html#K1" title="JORDAN2B:func.1">|[</a><span class="default"><span class="p2">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p3">(<span class="default"><span class="p4"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><font color="Maroon" title="c8">r</font></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p4"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><font color="Maroon" title="c3">r</font></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span></span>)</span></span>)</span></span><a href="jordan2b.html#K1" title="JORDAN2B:func.1">]|</a></span> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c7">P1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:114_1_1_2_1_2"><span class="lab"><font color="Green" title="E91">A18</font></span></a></span>;<br/><span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><span class="p2"><a href="jordan2b.html#K1" title="JORDAN2B:func.1">|[</a><span class="default"><span class="p3">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p4">(<span class="default"><span class="p5"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><font color="Maroon" title="c8">r</font></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p5"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><font color="Maroon" title="c3">r</font></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span></span>)</span></span>)</span></span><a href="jordan2b.html#K1" title="JORDAN2B:func.1">]|</a></span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> = 
<span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p3">(<span class="default"><span class="p4"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><font color="Maroon" title="c8">r</font></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p4"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><font color="Maroon" title="c3">r</font></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span></span>)</span></span>)</span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="jordan2c.html#T10" target="_self" onmouseover="rs('jordan2c/T10')" onmouseout="rh()">Th4</a></span>
<br/>.= 
<span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><font color="Maroon" title="c8">r</font></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p3"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><font color="Maroon" title="c3">r</font></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span></span>)</span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="complex1.html#T52" onmouseover="rs('complex1/T52')" onmouseout="rh()">COMPLEX1:52</a></span>
;<br/><span class="kw">hence </span><a NAME="E10:114_1_1_2_1_2_7"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:114_1_1_2_1_2_7"><span class="lab"><font color="Green" title="E99">A58</font></span></a>, <a class="txt" href="#E8:114_1_1_2_1_2_7"><span class="lab"><font color="Green" title="E102">A63</font></span></a>, <a class="txt" href="#E5:114_1_1_2_1_2_7"><span class="lab"><font color="Green" title="E100">A59</font></span></a>, <a class="txt" href="#E6:114_1_1_2_1_2_7"><span class="lab"><font color="Green" title="E101">A60</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><a NAME="E21:114_1_1_2_1_2"/>
 for <font color="Olive" title="b1">w1</font>, <font color="Olive" title="b2">w2</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">n</font></span>)</span>  st <font color="Olive" title="b1">w1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c7">P1</font> &amp; <font color="Olive" title="b2">w2</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c7">P1</font> holds <br/> <a href="rltopsp1.html#K1" title="RLTOPSP1:func.1">LSeg</a> (<font color="Olive" title="b1">w1</font>,<font color="Olive" title="b2">w2</font>) <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c7">P1</font>
 <div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="114_1_1_2_1_2_8"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c8">w1</font>, <font color="Maroon" title="c9">w2</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">n</font></span>)</span>; <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="c8">w1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c7">P1</font> &amp; <font color="Maroon" title="c9">w2</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c7">P1</font> implies  <a href="rltopsp1.html#K1" title="RLTOPSP1:func.1">LSeg</a> (<font color="Maroon" title="c8">w1</font>,<font color="Maroon" title="c9">w2</font>) <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c7">P1</font> )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:114_1_1_2_1_2_8"/><span class="lab"><font color="Green" title="E99">A64</font></span>: 
<font color="Maroon" title="c8">w1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c7">P1</font>
 <span class="kw">and </span><br/><a NAME="E2:114_1_1_2_1_2_8"/><span class="lab"><font color="Green" title="E100">A65</font></span>: 
<font color="Maroon" title="c9">w2</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c7">P1</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">  <a href="rltopsp1.html#K1" title="RLTOPSP1:func.1">LSeg</a> (<font color="Maroon" title="c8">w1</font>,<font color="Maroon" title="c9">w2</font>) <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c7">P1</font></span><br/>

<span class="kw">consider </span><font color="Maroon" title="c10">r4</font> being   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a><span class="kw"> such that </span><br/><a NAME="E4:114_1_1_2_1_2_8"/><span class="lab"><font color="Green" title="E101">A66</font></span>: 
<font color="Maroon" title="c9">w2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><font color="Maroon" title="c10">r4</font></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:114_1_1_2_1_2"><span class="lab"><font color="Green" title="E91">A18</font></span></a>, <a class="ref" href="jordan2b.html#T20" onmouseover="rs('jordan2b/T20')" onmouseout="rh()">JORDAN2B:20</a></span>;<br/>
<a NAME="E5:114_1_1_2_1_2_8"/>
 ex <font color="Olive" title="b1">q2</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">n</font></span>)</span> st <br/>( <font color="Olive" title="b1">q2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">w2</font> &amp; (  for <font color="Olive" title="b2">r2</font> being   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a>  st <font color="Olive" title="b1">q2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="jordan2b.html#K1" title="JORDAN2B:func.1">|[</a><span class="default"><font color="Olive" title="b2">r2</font></span><a href="jordan2b.html#K1" title="JORDAN2B:func.1">]|</a></span> holds <br/><font color="Olive" title="b2">r2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <font color="Maroon" title="c3">r</font> ) )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:114_1_1_2_1_2_8"><span class="lab"><font color="Green" title="E100">A65</font></span></a></span>;<br/>
<a NAME="E6:114_1_1_2_1_2_8"/><span class="kw">then </span><span class="lab"><font color="Green" title="E102">A67</font></span>: 
<font color="Maroon" title="c10">r4</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <font color="Maroon" title="c3">r</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:114_1_1_2_1_2_8"><span class="lab"><font color="Green" title="E101">A66</font></span></a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c11">r3</font> being   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a><span class="kw"> such that </span><br/><a NAME="E8:114_1_1_2_1_2_8"/><span class="lab"><font color="Green" title="E103">A68</font></span>: 
<font color="Maroon" title="c8">w1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><font color="Maroon" title="c11">r3</font></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:114_1_1_2_1_2"><span class="lab"><font color="Green" title="E91">A18</font></span></a>, <a class="ref" href="jordan2b.html#T20" onmouseover="rs('jordan2b/T20')" onmouseout="rh()">JORDAN2B:20</a></span>;<br/>
<a NAME="E9:114_1_1_2_1_2_8"/>
 ex <font color="Olive" title="b1">q1</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">n</font></span>)</span> st <br/>( <font color="Olive" title="b1">q1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">w1</font> &amp; (  for <font color="Olive" title="b2">r2</font> being   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a>  st <font color="Olive" title="b1">q1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="jordan2b.html#K1" title="JORDAN2B:func.1">|[</a><span class="default"><font color="Olive" title="b2">r2</font></span><a href="jordan2b.html#K1" title="JORDAN2B:func.1">]|</a></span> holds <br/><font color="Olive" title="b2">r2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <font color="Maroon" title="c3">r</font> ) )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:114_1_1_2_1_2_8"><span class="lab"><font color="Green" title="E99">A64</font></span></a></span>;<br/>
<a NAME="E10:114_1_1_2_1_2_8"/><span class="kw">then </span><span class="lab"><font color="Green" title="E104">A69</font></span>: 
<font color="Maroon" title="c11">r3</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <font color="Maroon" title="c3">r</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:114_1_1_2_1_2_8"><span class="lab"><font color="Green" title="E103">A68</font></span></a></span>;<br/>
<span class="kw">thus </span><a NAME="E11:114_1_1_2_1_2_8"/>
 <a href="rltopsp1.html#K1" title="RLTOPSP1:func.1">LSeg</a> (<font color="Maroon" title="c8">w1</font>,<font color="Maroon" title="c9">w2</font>) <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c7">P1</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><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="114_1_1_2_1_2_8_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c12">z</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="c12">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rltopsp1.html#K1" title="RLTOPSP1:func.1">LSeg</a> (<font color="Maroon" title="c8">w1</font>,<font color="Maroon" title="c9">w2</font>) or <font color="Maroon" title="c12">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">P1</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:114_1_1_2_1_2_8_1"/>
<font color="Maroon" title="c12">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rltopsp1.html#K1" title="RLTOPSP1:func.1">LSeg</a> (<font color="Maroon" title="c8">w1</font>,<font color="Maroon" title="c9">w2</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="c12">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">P1</font></span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c13">r2</font> being   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a><span class="kw"> such that </span><br/><a NAME="E3:114_1_1_2_1_2_8_1"/><span class="lab"><font color="Green" title="E105">A70</font></span>: 
<font color="Maroon" title="c12">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c13">r2</font></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c8">w1</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">r2</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c9">w2</font></span>)</span>
 <span class="kw">and </span><br/><a NAME="E4:114_1_1_2_1_2_8_1"/><span class="lab"><font color="Green" title="E106">A71</font></span>: 
 <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="c13">r2</font>
 <span class="kw">and </span><br/><a NAME="E5:114_1_1_2_1_2_8_1"/><span class="lab"><font color="Green" title="E107">A72</font></span>: 
<font color="Maroon" title="c13">r2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1
 ;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c14">q4</font> = <font color="Maroon" title="c12">z</font> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">n</font></span>)</span> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:114_1_1_2_1_2_8_1"><span class="lab"><font color="Green" title="E105">A70</font></span></a></span>;<br/>
<a NAME="E7:114_1_1_2_1_2_8_1"/><span class="lab"><font color="Green" title="E108">A73</font></span>: 
<font color="Maroon" title="c13">r2</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c9">w2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="jordan2b.html#K1" title="JORDAN2B:func.1">|[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c13">r2</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c10">r4</font></span>)</span></span><a href="jordan2b.html#K1" title="JORDAN2B:func.1">]|</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:114_1_1_2_1_2"><span class="lab"><font color="Green" title="E91">A18</font></span></a>, <a class="txt" href="#E4:114_1_1_2_1_2_8"><span class="lab"><font color="Green" title="E101">A66</font></span></a>, <a class="ref" href="jordan2b.html#T21" onmouseover="rs('jordan2b/T21')" onmouseout="rh()">JORDAN2B:21</a></span>;<br/>
<span class="p1">(<span class="default">1 <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c13">r2</font></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c8">w1</font> = 
<span class="p1">(<span class="default">1 <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c13">r2</font></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <span class="p1"><a href="jordan2b.html#K1" title="JORDAN2B:func.1">|[</a><span class="default"><font color="Maroon" title="c11">r3</font></span><a href="jordan2b.html#K1" title="JORDAN2B:func.1">]|</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:114_1_1_2_1_2_8"><span class="lab"><font color="Green" title="E103">A68</font></span></a></span>
<br/>.= 
<span class="p1"><a href="jordan2b.html#K1" title="JORDAN2B:func.1">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c13">r2</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c11">r3</font></span>)</span></span><a href="jordan2b.html#K1" title="JORDAN2B:func.1">]|</a></span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="jordan2b.html#T21" onmouseover="rs('jordan2b/T21')" onmouseout="rh()">JORDAN2B:21</a></span>
;<br/>
<a NAME="E9:114_1_1_2_1_2_8_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E109">A74</font></span>: 
<font color="Maroon" title="c12">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="jordan2b.html#K1" title="JORDAN2B:func.1">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default">1 <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c13">r2</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c11">r3</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c13">r2</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c10">r4</font></span>)</span></span>)</span></span><a href="jordan2b.html#K1" title="JORDAN2B:func.1">]|</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:114_1_1_2_1_2"><span class="lab"><font color="Green" title="E91">A18</font></span></a>, <a class="txt" href="#E3:114_1_1_2_1_2_8_1"><span class="lab"><font color="Green" title="E105">A70</font></span></a>, <a class="txt" href="#E7:114_1_1_2_1_2_8_1"><span class="lab"><font color="Green" title="E108">A73</font></span></a>, <a class="ref" href="jordan2b.html#T22" onmouseover="rs('jordan2b/T22')" onmouseout="rh()">JORDAN2B:22</a></span>;<br/>
<a NAME="E10:114_1_1_2_1_2_8_1"/>
 for <font color="Olive" title="b1">r5</font> being   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a>  st <font color="Maroon" title="c14">q4</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="jordan2b.html#K1" title="JORDAN2B:func.1">|[</a><span class="default"><font color="Olive" title="b1">r5</font></span><a href="jordan2b.html#K1" title="JORDAN2B:func.1">]|</a></span> holds <br/><font color="Olive" title="b1">r5</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <font color="Maroon" title="c3">r</font>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="114_1_1_2_1_2_8_1_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c15">r5</font> be   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</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="c14">q4</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="jordan2b.html#K1" title="JORDAN2B:func.1">|[</a><span class="default"><font color="Maroon" title="c15">r5</font></span><a href="jordan2b.html#K1" title="JORDAN2B:func.1">]|</a></span> implies <font color="Maroon" title="c15">r5</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <font color="Maroon" title="c3">r</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:114_1_1_2_1_2_8_1_2"/>
<font color="Maroon" title="c14">q4</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="jordan2b.html#K1" title="JORDAN2B:func.1">|[</a><span class="default"><font color="Maroon" title="c15">r5</font></span><a href="jordan2b.html#K1" title="JORDAN2B:func.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"> <font color="Maroon" title="c15">r5</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <font color="Maroon" title="c3">r</font></span><br/>

<a NAME="E2:114_1_1_2_1_2_8_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E110">A75</font></span>: 
<font color="Maroon" title="c15">r5</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c13">r2</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c11">r3</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">r2</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c10">r4</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:114_1_1_2_1_2_8_1"><span class="lab"><font color="Green" title="E109">A74</font></span></a>, <a class="ref" href="jordan2b.html#T23" onmouseover="rs('jordan2b/T23')" onmouseout="rh()">JORDAN2B:23</a></span>;<br/>
<a NAME="E3:114_1_1_2_1_2_8_1_2"/>
1 <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c13">r2</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="#E5:114_1_1_2_1_2_8_1"><span class="lab"><font color="Green" title="E107">A72</font></span></a>, <a class="ref" href="xreal_1.html#T48" onmouseover="rs('xreal_1/T48')" onmouseout="rh()">XREAL_1:48</a></span>;<br/>
<a NAME="E4:114_1_1_2_1_2_8_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E111">A76</font></span>: 
<span class="p1">(<span class="default">1 <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c13">r2</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c11">r3</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default">1 <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c13">r2</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <font color="Maroon" title="c3">r</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:114_1_1_2_1_2_8"><span class="lab"><font color="Green" title="E104">A69</font></span></a>, <a class="ref" href="xreal_1.html#T64" onmouseover="rs('xreal_1/T64')" onmouseout="rh()">XREAL_1:64</a></span>;<br/>
<a NAME="E5:114_1_1_2_1_2_8_1_2"/>
( <font color="Maroon" title="c13">r2</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c10">r4</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c13">r2</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <font color="Maroon" title="c3">r</font></span>)</span> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c13">r2</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <font color="Maroon" title="c3">r</font></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="c13">r2</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <font color="Maroon" title="c3">r</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <font color="Maroon" title="c3">r</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:114_1_1_2_1_2_8"><span class="lab"><font color="Green" title="E102">A67</font></span></a>, <a class="txt" href="#E4:114_1_1_2_1_2_8_1"><span class="lab"><font color="Green" title="E106">A71</font></span></a>, <a class="ref" href="xreal_1.html#T64" onmouseover="rs('xreal_1/T64')" onmouseout="rh()">XREAL_1:64</a></span>;<br/>
<span class="kw">hence </span><a NAME="E6:114_1_1_2_1_2_8_1_2"/>
<font color="Maroon" title="c15">r5</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <font color="Maroon" title="c3">r</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:114_1_1_2_1_2_8_1_2"><span class="lab"><font color="Green" title="E110">A75</font></span></a>, <a class="txt" href="#E4:114_1_1_2_1_2_8_1_2"><span class="lab"><font color="Green" title="E111">A76</font></span></a>, <a class="ref" href="xreal_1.html#T7" onmouseover="rs('xreal_1/T7')" onmouseout="rh()">XREAL_1:7</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="E11:114_1_1_2_1_2_8_1"/>
<font color="Maroon" title="c12">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">P1</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>


</div><span class="kw">end;</span></div><a NAME="E22:114_1_1_2_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c7">P1</font> is  <a href="convex1.html#V1" title="CONVEX1:attr.1">convex</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="jordan1.html#D1" onmouseover="rs('jordan1/D1')" onmouseout="rh()">JORDAN1:def 1</a></span>;<br/><a NAME="E23:114_1_1_2_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E99">A77</font></span>: 
 <a href="connsp_3.html#K4" title="CONNSP_3:func.4">Down</a> (<font color="Maroon" title="c7">P1</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a></span>)</span>) is  <a href="connsp_1.html#V2" title="CONNSP_1:attr.2">connected</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:114_1_1_2_1_2"><span class="lab"><font color="Green" title="E96">A50</font></span></a>, <a class="ref" href="connsp_1.html#T46" onmouseover="rs('connsp_1/T46')" onmouseout="rh()">CONNSP_1:46</a></span>;<br/><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="114_1_1_2_1_2_9"><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="jordan2c.html#K1" title="JORDAN2C:func.1">BDD</a> <font color="Maroon" title="c2">A</font> is  <a href="rltopsp1.html#V9" title="RLTOPSP1:attr.9">bounded</a> </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:114_1_1_2_1_2_9"/>
 not  <a href="jordan2c.html#K1" title="JORDAN2C:func.1">BDD</a> <font color="Maroon" title="c2">A</font> is  <a href="rltopsp1.html#V9" title="RLTOPSP1:attr.9">bounded</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/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c8">q</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">n</font></span>)</span><span class="kw"> such that </span><br/><a NAME="E3:114_1_1_2_1_2_9"/><span class="lab"><font color="Green" title="E100">A78</font></span>: 
<font color="Maroon" title="c8">q</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="jordan2c.html#K1" title="JORDAN2C:func.1">BDD</a> <font color="Maroon" title="c2">A</font>
 <span class="kw">and </span><br/><a NAME="E4:114_1_1_2_1_2_9"/><span class="lab"><font color="Green" title="E101">A79</font></span>: 
 not <span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><font color="Maroon" title="c8">q</font></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">r</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="jordan2c.html#T34" target="_self" onmouseover="rs('jordan2c/T34')" onmouseout="rh()">Th21</a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c9">y</font> being    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a> <span class="kw"> such that </span><br/><a NAME="E6:114_1_1_2_1_2_9"/><span class="lab"><font color="Green" title="E102">A80</font></span>: 
<font color="Maroon" title="c8">q</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c9">y</font>
 <span class="kw">and </span><br/><a NAME="E7:114_1_1_2_1_2_9"/><span class="lab"><font color="Green" title="E103">A81</font></span>: 
<font color="Maroon" title="c9">y</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">B3</font> where <font color="Olive" title="b1">B3</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">n</font></span>)</span> : <font color="Olive" title="b1">B3</font> <a href="jordan2c.html#R1" title="JORDAN2C:pred.1">is_inside_component_of</a> <font color="Maroon" title="c2">A</font> </span> } </span> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:114_1_1_2_1_2_9"><span class="lab"><font color="Green" title="E100">A78</font></span></a>, <a class="ref" href="tarski.html#D4" onmouseover="rs('tarski/D4')" onmouseout="rh()">TARSKI:def 4</a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c10">B3</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">n</font></span>)</span><span class="kw"> such that </span><br/><a NAME="E9:114_1_1_2_1_2_9"/><span class="lab"><font color="Green" title="E104">A82</font></span>: 
<font color="Maroon" title="c9">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">B3</font>
 <span class="kw">and </span><br/><a NAME="E10:114_1_1_2_1_2_9"/><span class="lab"><font color="Green" title="E105">A83</font></span>: 
<font color="Maroon" title="c10">B3</font> <a href="jordan2c.html#R1" title="JORDAN2C:pred.1">is_inside_component_of</a> <font color="Maroon" title="c2">A</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:114_1_1_2_1_2_9"><span class="lab"><font color="Green" title="E103">A81</font></span></a></span>;<br/><a NAME="E11:114_1_1_2_1_2_9"/>
<font color="Maroon" title="c8">q</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 <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">n</font></span>)</span>
 ;<br/><a NAME="E12:114_1_1_2_1_2_9"/><span class="kw">then </span><span class="lab"><font color="Green" title="E106">A84</font></span>: 
<font color="Maroon" title="c8">q</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="euclid.html#K1" title="EUCLID:func.1">REAL</a> <font color="Maroon" title="c1">n</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="euclid.html#T22" onmouseover="rs('euclid/T22')" onmouseout="rh()">EUCLID:22</a></span>;<br/><a NAME="E13:114_1_1_2_1_2_9"/>
 for <font color="Olive" title="b1">q2</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">n</font></span>)</span>  st <font color="Olive" title="b1">q2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">q</font> holds <br/><span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><font color="Olive" title="b1">q2</font></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c3">r</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:114_1_1_2_1_2_9"><span class="lab"><font color="Green" title="E101">A79</font></span></a></span>;<br/><a NAME="E14:114_1_1_2_1_2_9"/><span class="kw">then </span>
 not <font color="Maroon" title="c8">q</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">q2</font> where <font color="Olive" title="b1">q2</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">n</font></span>)</span> : <span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><font color="Olive" title="b1">q2</font></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">r</font> </span> } </span> 
 ;<br/><a NAME="E15:114_1_1_2_1_2_9"/><span class="kw">then </span><span class="lab"><font color="Green" title="E107">A85</font></span>: 
<font color="Maroon" title="c8">q</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c5">P</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:114_1_1_2_1_2_9"><span class="lab"><font color="Green" title="E106">A84</font></span></a>, <a class="ref" href="xboole_0.html#D5" onmouseover="rs('xboole_0/D5')" onmouseout="rh()">XBOOLE_0:def 5</a></span>;<br/><a NAME="E16:114_1_1_2_1_2_9"/>
<font color="Maroon" title="c10">B3</font> <a href="connsp_1.html#R3" title="CONNSP_1:pred.3">is_a_component_of</a> <font color="Maroon" title="c2">A</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:114_1_1_2_1_2_9"><span class="lab"><font color="Green" title="E105">A83</font></span></a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c11">B4</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a></span>)</span></span>)</span><span class="kw"> such that </span><br/><a NAME="E18:114_1_1_2_1_2_9"/><span class="lab"><font color="Green" title="E108">A86</font></span>: 
<font color="Maroon" title="c11">B4</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">B3</font>
 <span class="kw">and </span><br/><a NAME="E19:114_1_1_2_1_2_9"/><span class="lab"><font color="Green" title="E109">A87</font></span>: 
<font color="Maroon" title="c11">B4</font> is  <a href="connsp_1.html#V3" title="CONNSP_1:attr.3">a_component</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="connsp_1.html#D6" onmouseover="rs('connsp_1/D6')" onmouseout="rh()">CONNSP_1:def 6</a></span>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="114_1_1_2_1_2_9_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="E1:114_1_1_2_1_2_9_1"/>
( <font color="Maroon" title="c8">q</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c7">P1</font> or <font color="Maroon" title="c8">q</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c6">P2</font> )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:114_1_1_2_1_2"><span class="lab"><font color="Green" title="E92">A19</font></span></a>, <a class="txt" href="#E15:114_1_1_2_1_2_9"><span class="lab"><font color="Green" title="E107">A85</font></span></a>, <a class="ref" href="xboole_0.html#D3" onmouseover="rs('xboole_0/D3')" onmouseout="rh()">XBOOLE_0:def 3</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="114_1_1_2_1_2_9_1_1"><span class="kw">suppose </span></a><a NAME="E1:114_1_1_2_1_2_9_1_1"/>
<font color="Maroon" title="c8">q</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c7">P1</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"> contradiction</span><br/><div class="add"><a NAME="E2:114_1_1_2_1_2_9_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c7">P1</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c11">B4</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="struct_0.html#K1" title="STRUCT_0:func.1">{}</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:114_1_1_2_1_2_9"><span class="lab"><font color="Green" title="E102">A80</font></span></a>, <a class="txt" href="#E9:114_1_1_2_1_2_9"><span class="lab"><font color="Green" title="E104">A82</font></span></a>, <a class="txt" href="#E18:114_1_1_2_1_2_9"><span class="lab"><font color="Green" title="E108">A86</font></span></a>, <a class="ref" href="xboole_0.html#D4" onmouseover="rs('xboole_0/D4')" onmouseout="rh()">XBOOLE_0:def 4</a></span>;<br/><a NAME="E3:114_1_1_2_1_2_9_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E110">A88</font></span>: 
<font color="Maroon" title="c7">P1</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c11">B4</font>
 ;<br/><a NAME="E4:114_1_1_2_1_2_9_1_1"/>
<font color="Maroon" title="c10">B3</font> is  <a href="rltopsp1.html#V9" title="RLTOPSP1:attr.9">bounded</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:114_1_1_2_1_2_9"><span class="lab"><font color="Green" title="E105">A83</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E5:114_1_1_2_1_2_9_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:114_1_1_2_1_2"><span class="lab"><font color="Green" title="E96">A50</font></span></a>, <a class="txt" href="#E20:114_1_1_2_1_2"><span class="lab"><font color="Green" title="E98">A57</font></span></a>, <a class="txt" href="#E23:114_1_1_2_1_2"><span class="lab"><font color="Green" title="E99">A77</font></span></a>, <a class="txt" href="#E18:114_1_1_2_1_2_9"><span class="lab"><font color="Green" title="E108">A86</font></span></a>, <a class="txt" href="#E19:114_1_1_2_1_2_9"><span class="lab"><font color="Green" title="E109">A87</font></span></a>, <a class="txt" href="#E3:114_1_1_2_1_2_9_1_1"><span class="lab"><font color="Green" title="E110">A88</font></span></a>, <a class="ref" href="connsp_1.html#T36" onmouseover="rs('connsp_1/T36')" onmouseout="rh()">CONNSP_1:36</a>, <a class="ref" href="rltopsp1.html#T42" onmouseover="rs('rltopsp1/T42')" onmouseout="rh()">RLTOPSP1:42</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="114_1_1_2_1_2_9_1_2"><span class="kw">suppose </span></a><a NAME="E1:114_1_1_2_1_2_9_1_2"/>
<font color="Maroon" title="c8">q</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c6">P2</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"> contradiction</span><br/><div class="add"><a NAME="E2:114_1_1_2_1_2_9_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c6">P2</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c11">B4</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="struct_0.html#K1" title="STRUCT_0:func.1">{}</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:114_1_1_2_1_2_9"><span class="lab"><font color="Green" title="E102">A80</font></span></a>, <a class="txt" href="#E9:114_1_1_2_1_2_9"><span class="lab"><font color="Green" title="E104">A82</font></span></a>, <a class="txt" href="#E18:114_1_1_2_1_2_9"><span class="lab"><font color="Green" title="E108">A86</font></span></a>, <a class="ref" href="xboole_0.html#D4" onmouseover="rs('xboole_0/D4')" onmouseout="rh()">XBOOLE_0:def 4</a></span>;<br/><a NAME="E3:114_1_1_2_1_2_9_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E110">A89</font></span>: 
<font color="Maroon" title="c6">P2</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c11">B4</font>
 ;<br/><a NAME="E4:114_1_1_2_1_2_9_1_2"/>
<font color="Maroon" title="c10">B3</font> is  <a href="rltopsp1.html#V9" title="RLTOPSP1:attr.9">bounded</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:114_1_1_2_1_2_9"><span class="lab"><font color="Green" title="E105">A83</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E5:114_1_1_2_1_2_9_1_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:114_1_1_2_1_2"><span class="lab"><font color="Green" title="E94">A36</font></span></a>, <a class="txt" href="#E19:114_1_1_2_1_2"><span class="lab"><font color="Green" title="E97">A51</font></span></a>, <a class="txt" href="#E16:114_1_1_2_1_2"><span class="lab"><font color="Green" title="E95">A49</font></span></a>, <a class="txt" href="#E18:114_1_1_2_1_2_9"><span class="lab"><font color="Green" title="E108">A86</font></span></a>, <a class="txt" href="#E19:114_1_1_2_1_2_9"><span class="lab"><font color="Green" title="E109">A87</font></span></a>, <a class="txt" href="#E3:114_1_1_2_1_2_9_1_2"><span class="lab"><font color="Green" title="E110">A89</font></span></a>, <a class="ref" href="connsp_1.html#T36" onmouseover="rs('connsp_1/T36')" onmouseout="rh()">CONNSP_1:36</a>, <a class="ref" href="rltopsp1.html#T42" onmouseover="rs('rltopsp1/T42')" onmouseout="rh()">RLTOPSP1:42</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><span class="kw">hence </span><a NAME="E25:114_1_1_2_1_2"/>
 <a href="jordan2c.html#K1" title="JORDAN2C:func.1">BDD</a> <font color="Maroon" title="c2">A</font> is  <a href="rltopsp1.html#V9" title="RLTOPSP1:attr.9">bounded</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"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E6:114_1_1"/>
 <a href="jordan2c.html#K1" title="JORDAN2C:func.1">BDD</a> <font color="Maroon" title="c2">A</font> is  <a href="rltopsp1.html#V9" title="RLTOPSP1:attr.9">bounded</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"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="114_1_2"><span class="kw">suppose </span></a><a NAME="E1:114_1_2"/>
<font color="Maroon" title="c1">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1
 ; <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="jordan2c.html#K1" title="JORDAN2C:func.1">BDD</a> <font color="Maroon" title="c2">A</font> is  <a href="rltopsp1.html#V9" title="RLTOPSP1:attr.9">bounded</a> </span><br/><div class="add"><a NAME="E2:114_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c1">n</font> <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 href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 ;<br/><a NAME="E3:114_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E87">A90</font></span>: 
<font color="Maroon" title="c1">n</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="ref" href="nat_1.html#T13" onmouseover="rs('nat_1/T13')" onmouseout="rh()">NAT_1:13</a></span>;<br/><a NAME="E4:114_1_2"/>
 for <font color="Olive" title="b1">q2</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">n</font></span>)</span>  holds <span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><font color="Olive" title="b1">q2</font></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1
 <div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="114_1_2_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c4">q2</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">n</font></span>)</span>; <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"> <span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><font color="Maroon" title="c4">q2</font></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1</span><br/>

<a NAME="E1:114_1_2_1"/>
<font color="Maroon" title="c4">q2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">n</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:114_1_2"><span class="lab"><font color="Green" title="E87">A90</font></span></a>, <a class="ref" href="euclid.html#T77" onmouseover="rs('euclid/T77')" onmouseout="rh()">EUCLID:77</a></span>;<br/>
<span class="kw">hence </span><a NAME="E2:114_1_2_1"/>
<span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><font color="Maroon" title="c4">q2</font></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="toprns_1.html#T23" onmouseover="rs('toprns_1/T23')" onmouseout="rh()">TOPRNS_1:23</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="E5:114_1_2"/><span class="kw">then </span>
 for <font color="Olive" title="b1">q2</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">n</font></span>)</span>  st <font color="Olive" title="b1">q2</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">n</font></span>)</span> holds <br/><span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><font color="Olive" title="b1">q2</font></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1
 ;<br/><a NAME="E6:114_1_2"/><span class="kw">then </span>
 <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">n</font></span>)</span> is  <a href="rltopsp1.html#V9" title="RLTOPSP1:attr.9">bounded</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="jordan2c.html#T34" target="_self" onmouseover="rs('jordan2c/T34')" onmouseout="rh()">Th21</a></span>;<br/><span class="kw">hence </span><a NAME="E7:114_1_2"/>
 <a href="jordan2c.html#K1" title="JORDAN2C:func.1">BDD</a> <font color="Maroon" title="c2">A</font> is  <a href="rltopsp1.html#V9" title="RLTOPSP1:attr.9">bounded</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="rltopsp1.html#T42" onmouseover="rs('rltopsp1/T42')" onmouseout="rh()">RLTOPSP1:42</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>
