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<b>let </b><font color="Maroon" title="c2">n</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">r</font> being non <a href="xxreal_0.html#V3" title="XXREAL_0:attr.3">negative</a> <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> <br/> for <font color="Olive" title="b2">s</font>, <font color="Olive" title="b3">t</font>, <font color="Olive" title="b4">x</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="c2">n</font></span>)</span>  st <font color="Olive" title="b2">s</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b3">t</font> &amp; <font color="Olive" title="b2">s</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="brouwer.html#K2" title="BROUWER:func.2">Tdisk</a> <font color="Olive" title="b4">x</font>,<font color="Olive" title="b1">r</font></span>)</span> &amp; <font color="Olive" title="b2">s</font> is  not   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="toprealb.html#K7" title="TOPREALB:func.7">Tcircle</a> <font color="Olive" title="b4">x</font>,<font color="Olive" title="b1">r</font></span>)</span> holds <br/> ex <font color="Olive" title="b5">e</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="toprealb.html#K7" title="TOPREALB:func.7">Tcircle</a> <font color="Olive" title="b4">x</font>,<font color="Olive" title="b1">r</font></span>)</span> st <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Olive" title="b5">e</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="topreal9.html#K4" title="TOPREAL9:func.4">halfline</a> <font color="Olive" title="b2">s</font>,<font color="Olive" title="b3">t</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal9.html#K3" title="TOPREAL9:func.3">Sphere</a> <font color="Olive" title="b4">x</font>,<font color="Olive" title="b1">r</font></span>)</span></span><br/><b>let </b><font color="Maroon" title="c3">r</font> be  non <a href="xxreal_0.html#V3" title="XXREAL_0:attr.3">negative</a> <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">s</font>, <font color="Olive" title="b2">t</font>, <font color="Olive" title="b3">x</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="c2">n</font></span>)</span>  st <font color="Olive" title="b1">s</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b2">t</font> &amp; <font color="Olive" title="b1">s</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="brouwer.html#K2" title="BROUWER:func.2">Tdisk</a> <font color="Olive" title="b3">x</font>,<font color="Maroon" title="c3">r</font></span>)</span> &amp; <font color="Olive" title="b1">s</font> is  not   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="toprealb.html#K7" title="TOPREALB:func.7">Tcircle</a> <font color="Olive" title="b3">x</font>,<font color="Maroon" title="c3">r</font></span>)</span> holds <br/> ex <font color="Olive" title="b4">e</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="toprealb.html#K7" title="TOPREALB:func.7">Tcircle</a> <font color="Olive" title="b3">x</font>,<font color="Maroon" title="c3">r</font></span>)</span> st <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Olive" title="b4">e</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="topreal9.html#K4" title="TOPREAL9:func.4">halfline</a> <font color="Olive" title="b1">s</font>,<font color="Olive" title="b2">t</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal9.html#K3" title="TOPREAL9:func.3">Sphere</a> <font color="Olive" title="b3">x</font>,<font color="Maroon" title="c3">r</font></span>)</span></span><br/><b>let </b><font color="Maroon" title="c4">s</font>, <font color="Maroon" title="c5">t</font>, <font color="Maroon" title="c6">x</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="c2">n</font></span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c4">s</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">t</font> &amp; <font color="Maroon" title="c4">s</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="brouwer.html#K2" title="BROUWER:func.2">Tdisk</a> <font color="Maroon" title="c6">x</font>,<font color="Maroon" title="c3">r</font></span>)</span> &amp; <font color="Maroon" title="c4">s</font> is  not   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="toprealb.html#K7" title="TOPREALB:func.7">Tcircle</a> <font color="Maroon" title="c6">x</font>,<font color="Maroon" title="c3">r</font></span>)</span> implies  ex <font color="Olive" title="b1">e</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="toprealb.html#K7" title="TOPREALB:func.7">Tcircle</a> <font color="Maroon" title="c6">x</font>,<font color="Maroon" title="c3">r</font></span>)</span> st <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Olive" title="b1">e</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="topreal9.html#K4" title="TOPREAL9:func.4">halfline</a> <font color="Maroon" title="c4">s</font>,<font color="Maroon" title="c5">t</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal9.html#K3" title="TOPREAL9:func.3">Sphere</a> <font color="Maroon" title="c6">x</font>,<font color="Maroon" title="c3">r</font></span>)</span> )</span><br/>









<b>assume </b><b>that </b><br/><a NAME="E1:9"/><i><font color="Green" title="E4">A1</font></i>: 
<font color="Maroon" title="c4">s</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">t</font>
 <b>and </b><br/><a NAME="E2:9"/><i><font color="Green" title="E5">A2</font></i>: 
<font color="Maroon" title="c4">s</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="brouwer.html#K2" title="BROUWER:func.2">Tdisk</a> <font color="Maroon" title="c6">x</font>,<font color="Maroon" title="c3">r</font></span>)</span>
 <b>and </b><br/><a NAME="E3:9"/><i><font color="Green" title="E6">A3</font></i>: 
<font color="Maroon" title="c4">s</font> is  not   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="toprealb.html#K7" title="TOPREALB:func.7">Tcircle</a> <font color="Maroon" title="c6">x</font>,<font color="Maroon" title="c3">r</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive" title="b1">e</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="toprealb.html#K7" title="TOPREALB:func.7">Tcircle</a> <font color="Maroon" title="c6">x</font>,<font color="Maroon" title="c3">r</font></span>)</span> st <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Olive" title="b1">e</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="topreal9.html#K4" title="TOPREAL9:func.4">halfline</a> <font color="Maroon" title="c4">s</font>,<font color="Maroon" title="c5">t</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal9.html#K3" title="TOPREAL9:func.3">Sphere</a> <font color="Maroon" title="c6">x</font>,<font color="Maroon" title="c3">r</font></span>)</span></span><br/>

<a NAME="E4:9"/><i><font color="Green" title="E7">A4</font></i>: 
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="toprealb.html#K7" title="TOPREALB:func.7">Tcircle</a> <font color="Maroon" title="c6">x</font>,<font color="Maroon" title="c3">r</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="topreal9.html#K3" title="TOPREAL9:func.3">Sphere</a> <font color="Maroon" title="c6">x</font>,<font color="Maroon" title="c3">r</font>
 
<b>by </b><i><a class="ref" href="toprealb.html#T9" title="TOPREALB:th.9">TOPREALB:9</a></i>;<br/>
<a NAME="E5:9"/><i><font color="Green" title="E8">A5</font></i>: 
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="brouwer.html#K2" title="BROUWER:func.2">Tdisk</a> <font color="Maroon" title="c6">x</font>,<font color="Maroon" title="c3">r</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="topreal9.html#K2" title="TOPREAL9:func.2">cl_Ball</a> <font color="Maroon" title="c6">x</font>,<font color="Maroon" title="c3">r</font>
 
<b>by </b><i><a class="ref" href="brouwer.html#T3" target="_self" title="BROUWER:th.3">Th3</a></i>;<br/>
<a NAME="E6:9"/><i><font color="Green" title="E9">A6</font></i>: 
<span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">s</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c6">x</font></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">r</font>
 
<b>by </b><i><a class="txt" href="brouwer.html#E3:9"><i><font color="Green" title="E6">A3</font></i></a>, <a class="txt" href="brouwer.html#E4:9"><i><font color="Green" title="E7">A4</font></i></a>, <a class="ref" href="topreal9.html#T9" title="TOPREAL9:th.9">TOPREAL9:9</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c7">S</font> = <font color="Maroon" title="c4">s</font>, <font color="Maroon" title="c8">T</font> = <font color="Maroon" title="c5">t</font>, <font color="Maroon" title="c9">X</font> = <font color="Maroon" title="c6">x</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="euclid.html#K1" title="EUCLID:func.1">REAL</a> <font color="Maroon" title="c2">n</font> <b>by </b><i><a class="ref" href="euclid.html#T25" title="EUCLID:th.25">EUCLID:25</a></i>;<br/>
<b>set </b><font color="Maroon" title="c10">a</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p3">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p4"><a href="euclid_2.html#K2" title="EUCLID_2:func.2">|(</a><span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c5">t</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c4">s</font></span>)</span>,<span class="p5">(<span class="default"><font color="Maroon" title="c4">s</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c6">x</font></span>)</span></span><a href="euclid_2.html#K2" title="EUCLID_2:func.2">)|</a></span></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p2">(<span class="default"><a href="square_1.html#K7" title="SQUARE_1:func.7">sqrt</a> <span class="p3">(<span class="default"><a href="quin_1.html#K2" title="QUIN_1:func.2">delta</a> <span class="p4">(<span class="default"><a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p5">(<span class="default"><a href="euclid.html#K11" title="EUCLID:func.11">sqr</a> <span class="p0">(<span class="default"><font color="Maroon" title="c8">T</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c7">S</font></span>)</span></span>)</span></span>)</span>,<span class="p4">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p5"><a href="euclid_2.html#K2" title="EUCLID_2:func.2">|(</a><span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c5">t</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c4">s</font></span>)</span>,<span class="p0">(<span class="default"><font color="Maroon" title="c4">s</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c6">x</font></span>)</span></span><a href="euclid_2.html#K2" title="EUCLID_2:func.2">)|</a></span></span>)</span>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p0">(<span class="default"><a href="euclid.html#K11" title="EUCLID:func.11">sqr</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">S</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c9">X</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p5">(<span class="default"><font color="Maroon" title="c3">r</font> <a href="square_1.html#K3" title="SQUARE_1:func.3">^2</a> </span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2">(<span class="default"><a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p3">(<span class="default"><a href="euclid.html#K11" title="EUCLID:func.11">sqr</a> <span class="p4">(<span class="default"><font color="Maroon" title="c8">T</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c7">S</font></span>)</span></span>)</span></span>)</span></span>)</span>;<br/>
<a NAME="E8:9"/>
<span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">s</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c6">x</font></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">r</font>
 
<b>by </b><i><a class="txt" href="brouwer.html#E2:9"><i><font color="Green" title="E5">A2</font></i></a>, <a class="txt" href="brouwer.html#E5:9"><i><font color="Green" title="E8">A5</font></i></a>, <a class="ref" href="topreal9.html#T8" title="TOPREAL9:th.8">TOPREAL9:8</a></i>;<br/>
<a NAME="E9:9"/><b>then </b>
<span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">s</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c6">x</font></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">r</font>
 
<b>by </b><i><a class="txt" href="brouwer.html#E6:9"><i><font color="Green" title="E9">A6</font></i></a>, <a class="ref" href="real_1.html#D5" title="REAL_1:def.5">REAL_1:def 5</a></i>;<br/>
<a NAME="E10:9"/><b>then </b>
<font color="Maroon" title="c4">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal9.html#K1" title="TOPREAL9:func.1">Ball</a> <font color="Maroon" title="c6">x</font>,<font color="Maroon" title="c3">r</font>
 
<b>by </b><i><a class="ref" href="topreal9.html#T7" title="TOPREAL9:th.7">TOPREAL9:7</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c11">e1</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="c2">n</font></span>)</span><b> such that </b><br/><a NAME="E12:9"/><i><font color="Green" title="E10">A7</font></i>: 
<span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c11">e1</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="topreal9.html#K4" title="TOPREAL9:func.4">halfline</a> <font color="Maroon" title="c4">s</font>,<font color="Maroon" title="c5">t</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal9.html#K3" title="TOPREAL9:func.3">Sphere</a> <font color="Maroon" title="c6">x</font>,<font color="Maroon" title="c3">r</font></span>)</span>
 <b>and </b><br/><a NAME="E13:9"/>
<font color="Maroon" title="c11">e1</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="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p0">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1"><a href="euclid_2.html#K2" title="EUCLID_2:func.2">|(</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">t</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c4">s</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">s</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c6">x</font></span>)</span></span><a href="euclid_2.html#K2" title="EUCLID_2:func.2">)|</a></span></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p5">(<span class="default"><a href="square_1.html#K7" title="SQUARE_1:func.7">sqrt</a> <span class="p0">(<span class="default"><a href="quin_1.html#K2" title="QUIN_1:func.2">delta</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p2">(<span class="default"><a href="euclid.html#K11" title="EUCLID:func.11">sqr</a> <span class="p3">(<span class="default"><font color="Maroon" title="c8">T</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c7">S</font></span>)</span></span>)</span></span>)</span>,<span class="p1">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2"><a href="euclid_2.html#K2" title="EUCLID_2:func.2">|(</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c5">t</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c4">s</font></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon" title="c4">s</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c6">x</font></span>)</span></span><a href="euclid_2.html#K2" title="EUCLID_2:func.2">)|</a></span></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p3">(<span class="default"><a href="euclid.html#K11" title="EUCLID:func.11">sqr</a> <span class="p4">(<span class="default"><font color="Maroon" title="c7">S</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c9">X</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">r</font> <a href="square_1.html#K3" title="SQUARE_1:func.3">^2</a> </span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p4">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p5">(<span class="default"><a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p0">(<span class="default"><a href="euclid.html#K11" title="EUCLID:func.11">sqr</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">T</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c7">S</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <font color="Maroon" title="c4">s</font></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p5">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p0"><a href="euclid_2.html#K2" title="EUCLID_2:func.2">|(</a><span class="default"><span class="p1">(<span class="default"><font color="Maroon" title="c5">t</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c4">s</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">s</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c6">x</font></span>)</span></span><a href="euclid_2.html#K2" title="EUCLID_2:func.2">)|</a></span></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p4">(<span class="default"><a href="square_1.html#K7" title="SQUARE_1:func.7">sqrt</a> <span class="p5">(<span class="default"><a href="quin_1.html#K2" title="QUIN_1:func.2">delta</a> <span class="p0">(<span class="default"><a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p1">(<span class="default"><a href="euclid.html#K11" title="EUCLID:func.11">sqr</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">T</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c7">S</font></span>)</span></span>)</span></span>)</span>,<span class="p0">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1"><a href="euclid_2.html#K2" title="EUCLID_2:func.2">|(</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">t</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c4">s</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">s</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c6">x</font></span>)</span></span><a href="euclid_2.html#K2" title="EUCLID_2:func.2">)|</a></span></span>)</span>,<span class="p0">(<span class="default"><span class="p1">(<span class="default"><a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p2">(<span class="default"><a href="euclid.html#K11" title="EUCLID:func.11">sqr</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">S</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c9">X</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">r</font> <a href="square_1.html#K3" title="SQUARE_1:func.3">^2</a> </span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p3">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p4">(<span class="default"><a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p5">(<span class="default"><a href="euclid.html#K11" title="EUCLID:func.11">sqr</a> <span class="p0">(<span class="default"><font color="Maroon" title="c8">T</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c7">S</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <font color="Maroon" title="c5">t</font></span>)</span>
 <b>by </b><i><a class="txt" href="brouwer.html#E1:9"><i><font color="Green" title="E4">A1</font></i></a>, <a class="ref" href="topreal9.html#T37" title="TOPREAL9:th.37">TOPREAL9:37</a></i>;<br/>
<a NAME="E14:9"/>
<font color="Maroon" title="c11">e1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c11">e1</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<b>by </b><i><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/>
<a NAME="E15:9"/><b>then </b>
<font color="Maroon" title="c11">e1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal9.html#K3" title="TOPREAL9:func.3">Sphere</a> <font color="Maroon" title="c6">x</font>,<font color="Maroon" title="c3">r</font>
 
<b>by </b><i><a class="txt" href="brouwer.html#E12:9"><i><font color="Green" title="E10">A7</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/>
<b>hence </b><a NAME="E16:9"/>
 ex <font color="Olive" title="b1">e</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="toprealb.html#K7" title="TOPREALB:func.7">Tcircle</a> <font color="Maroon" title="c6">x</font>,<font color="Maroon" title="c3">r</font></span>)</span> st <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Olive" title="b1">e</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="topreal9.html#K4" title="TOPREAL9:func.4">halfline</a> <font color="Maroon" title="c4">s</font>,<font color="Maroon" title="c5">t</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal9.html#K3" title="TOPREAL9:func.3">Sphere</a> <font color="Maroon" title="c6">x</font>,<font color="Maroon" title="c3">r</font></span>)</span>
 <b>by </b><i><a class="txt" href="brouwer.html#E4:9"><i><font color="Green" title="E7">A4</font></i></a>, <a class="txt" href="brouwer.html#E12:9"><i><font color="Green" title="E10">A7</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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