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<span class="kw">let </span><font color="Maroon" title="c1">p</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> 2</span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c1">p</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  implies  for <font color="Olive" title="b1">x</font> being   <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">a</font> being   <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>   non  <a href="xxreal_0.html#V3" title="XXREAL_0:attr.3">negative</a>   <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> <br/>  for <font color="Olive" title="b3">y</font>, <font color="Olive" title="b4">r</font> being   <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>   <a href="xxreal_0.html#V2" title="XXREAL_0:attr.2">positive</a>   <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>   st <span class="p1">(<span class="default"><a href="topgen_5.html#K5" title="TOPGEN_5:func.5">+</a> (<font color="Olive" title="b1">x</font>,<font color="Olive" title="b3">y</font>,<font color="Olive" title="b4">r</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c1">p</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Olive" title="b2">a</font> holds <br/>( <span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><font color="Olive" title="b1">x</font>,<font color="Olive" title="b3">y</font></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <font color="Maroon" title="c1">p</font></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Olive" title="b4">r</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Olive" title="b2">a</font> &amp; <span class="p1">(<span class="default"><a href="topreal9.html#K1" title="TOPREAL9:func.1">Ball</a> (<font color="Maroon" title="c1">p</font>,<span class="p2">(<span class="default"><span class="p3"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><span class="p4">(<span class="default"><span class="p5"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><font color="Olive" title="b1">x</font>,<font color="Olive" title="b3">y</font></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <font color="Maroon" title="c1">p</font></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p3">(<span class="default"><font color="Olive" title="b4">r</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Olive" title="b2">a</font></span>)</span></span>)</span>)</span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <a href="topgen_5.html#K2" title="TOPGEN_5:func.2">y&gt;=0-plane</a> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><a href="topgen_5.html#K5" title="TOPGEN_5:func.5">+</a> (<font color="Olive" title="b1">x</font>,<font color="Olive" title="b3">y</font>,<font color="Olive" title="b4">r</font>)</span>)</span> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</a> <span class="p1"><a href="rcomp_1.html#K4" title="RCOMP_1:func.4">].</a><span class="default"><font color="Olive" title="b2">a</font>,1</span><a href="rcomp_1.html#K4" title="RCOMP_1:func.4">.]</a></span> ) )</span><br/>

<span class="kw">assume </span><a NAME="E1:88"/><span class="lab"><font color="Green" title="E70">A1</font></span>: 
<font color="Maroon" title="c1">p</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">x</font> being   <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">a</font> being   <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>   non  <a href="xxreal_0.html#V3" title="XXREAL_0:attr.3">negative</a>   <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> <br/>  for <font color="Olive" title="b3">y</font>, <font color="Olive" title="b4">r</font> being   <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>   <a href="xxreal_0.html#V2" title="XXREAL_0:attr.2">positive</a>   <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>   st <span class="p1">(<span class="default"><a href="topgen_5.html#K5" title="TOPGEN_5:func.5">+</a> (<font color="Olive" title="b1">x</font>,<font color="Olive" title="b3">y</font>,<font color="Olive" title="b4">r</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c1">p</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Olive" title="b2">a</font> holds <br/>( <span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><font color="Olive" title="b1">x</font>,<font color="Olive" title="b3">y</font></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <font color="Maroon" title="c1">p</font></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Olive" title="b4">r</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Olive" title="b2">a</font> &amp; <span class="p1">(<span class="default"><a href="topreal9.html#K1" title="TOPREAL9:func.1">Ball</a> (<font color="Maroon" title="c1">p</font>,<span class="p2">(<span class="default"><span class="p3"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><span class="p4">(<span class="default"><span class="p5"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><font color="Olive" title="b1">x</font>,<font color="Olive" title="b3">y</font></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <font color="Maroon" title="c1">p</font></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p3">(<span class="default"><font color="Olive" title="b4">r</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Olive" title="b2">a</font></span>)</span></span>)</span>)</span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <a href="topgen_5.html#K2" title="TOPGEN_5:func.2">y&gt;=0-plane</a> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><a href="topgen_5.html#K5" title="TOPGEN_5:func.5">+</a> (<font color="Olive" title="b1">x</font>,<font color="Olive" title="b3">y</font>,<font color="Olive" title="b4">r</font>)</span>)</span> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</a> <span class="p1"><a href="rcomp_1.html#K4" title="RCOMP_1:func.4">].</a><span class="default"><font color="Olive" title="b2">a</font>,1</span><a href="rcomp_1.html#K4" title="RCOMP_1:func.4">.]</a></span> )</span><br/>

<span class="kw">let </span><font color="Maroon" title="c2">x</font> be   <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>   <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">a</font> being   <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>   non  <a href="xxreal_0.html#V3" title="XXREAL_0:attr.3">negative</a>   <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> <br/>  for <font color="Olive" title="b2">y</font>, <font color="Olive" title="b3">r</font> being   <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>   <a href="xxreal_0.html#V2" title="XXREAL_0:attr.2">positive</a>   <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>   st <span class="p1">(<span class="default"><a href="topgen_5.html#K5" title="TOPGEN_5:func.5">+</a> (<font color="Maroon" title="c2">x</font>,<font color="Olive" title="b2">y</font>,<font color="Olive" title="b3">r</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c1">p</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Olive" title="b1">a</font> holds <br/>( <span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><font color="Maroon" title="c2">x</font>,<font color="Olive" title="b2">y</font></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <font color="Maroon" title="c1">p</font></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Olive" title="b3">r</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Olive" title="b1">a</font> &amp; <span class="p1">(<span class="default"><a href="topreal9.html#K1" title="TOPREAL9:func.1">Ball</a> (<font color="Maroon" title="c1">p</font>,<span class="p2">(<span class="default"><span class="p3"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><span class="p4">(<span class="default"><span class="p5"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><font color="Maroon" title="c2">x</font>,<font color="Olive" title="b2">y</font></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <font color="Maroon" title="c1">p</font></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p3">(<span class="default"><font color="Olive" title="b3">r</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Olive" title="b1">a</font></span>)</span></span>)</span>)</span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <a href="topgen_5.html#K2" title="TOPGEN_5:func.2">y&gt;=0-plane</a> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><a href="topgen_5.html#K5" title="TOPGEN_5:func.5">+</a> (<font color="Maroon" title="c2">x</font>,<font color="Olive" title="b2">y</font>,<font color="Olive" title="b3">r</font>)</span>)</span> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</a> <span class="p1"><a href="rcomp_1.html#K4" title="RCOMP_1:func.4">].</a><span class="default"><font color="Olive" title="b1">a</font>,1</span><a href="rcomp_1.html#K4" title="RCOMP_1:func.4">.]</a></span> )</span><br/><span class="kw">let </span><font color="Maroon" title="c3">a</font> be   <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>   non  <a href="xxreal_0.html#V3" title="XXREAL_0:attr.3">negative</a>   <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">y</font>, <font color="Olive" title="b2">r</font> being   <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>   <a href="xxreal_0.html#V2" title="XXREAL_0:attr.2">positive</a>   <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>   st <span class="p1">(<span class="default"><a href="topgen_5.html#K5" title="TOPGEN_5:func.5">+</a> (<font color="Maroon" title="c2">x</font>,<font color="Olive" title="b1">y</font>,<font color="Olive" title="b2">r</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c1">p</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c3">a</font> holds <br/>( <span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><font color="Maroon" title="c2">x</font>,<font color="Olive" title="b1">y</font></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <font color="Maroon" title="c1">p</font></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Olive" title="b2">r</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c3">a</font> &amp; <span class="p1">(<span class="default"><a href="topreal9.html#K1" title="TOPREAL9:func.1">Ball</a> (<font color="Maroon" title="c1">p</font>,<span class="p2">(<span class="default"><span class="p3"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><span class="p4">(<span class="default"><span class="p5"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><font color="Maroon" title="c2">x</font>,<font color="Olive" title="b1">y</font></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <font color="Maroon" title="c1">p</font></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p3">(<span class="default"><font color="Olive" title="b2">r</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c3">a</font></span>)</span></span>)</span>)</span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <a href="topgen_5.html#K2" title="TOPGEN_5:func.2">y&gt;=0-plane</a> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><a href="topgen_5.html#K5" title="TOPGEN_5:func.5">+</a> (<font color="Maroon" title="c2">x</font>,<font color="Olive" title="b1">y</font>,<font color="Olive" title="b2">r</font>)</span>)</span> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</a> <span class="p1"><a href="rcomp_1.html#K4" title="RCOMP_1:func.4">].</a><span class="default"><font color="Maroon" title="c3">a</font>,1</span><a href="rcomp_1.html#K4" title="RCOMP_1:func.4">.]</a></span> )</span><br/><span class="kw">let </span><font color="Maroon" title="c4">y</font>, <font color="Maroon" title="c5">r</font> be   <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>   <a href="xxreal_0.html#V2" title="XXREAL_0:attr.2">positive</a>   <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <span class="p1">(<span class="default"><a href="topgen_5.html#K5" title="TOPGEN_5:func.5">+</a> (<font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c4">y</font>,<font color="Maroon" title="c5">r</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c1">p</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c3">a</font> implies ( <span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c4">y</font></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <font color="Maroon" title="c1">p</font></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c5">r</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c3">a</font> &amp; <span class="p1">(<span class="default"><a href="topreal9.html#K1" title="TOPREAL9:func.1">Ball</a> (<font color="Maroon" title="c1">p</font>,<span class="p2">(<span class="default"><span class="p3"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><span class="p4">(<span class="default"><span class="p5"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c4">y</font></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <font color="Maroon" title="c1">p</font></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">r</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c3">a</font></span>)</span></span>)</span>)</span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <a href="topgen_5.html#K2" title="TOPGEN_5:func.2">y&gt;=0-plane</a> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><a href="topgen_5.html#K5" title="TOPGEN_5:func.5">+</a> (<font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c4">y</font>,<font color="Maroon" title="c5">r</font>)</span>)</span> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</a> <span class="p1"><a href="rcomp_1.html#K4" title="RCOMP_1:func.4">].</a><span class="default"><font color="Maroon" title="c3">a</font>,1</span><a href="rcomp_1.html#K4" title="RCOMP_1:func.4">.]</a></span> ) )</span><br/>





<span class="kw">set </span><font color="Maroon" title="c6">f</font> =  <a href="topgen_5.html#K5" title="TOPGEN_5:func.5">+</a> (<font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c4">y</font>,<font color="Maroon" title="c5">r</font>);<br/>
<a NAME="E2:88"/><span class="lab"><font color="Green" title="E71">A2</font></span>: 
<font color="Maroon" title="c1">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="euclid.html#T53" title="EUCLID:th.53">EUCLID:53</a></span>;<br/>
<a NAME="E3:88"/><span class="kw">then </span>
<font color="Maroon" title="c1">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topgen_5.html#K2" title="TOPGEN_5:func.2">y&gt;=0-plane</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:88"><span class="lab"><font color="Green" title="E70">A1</font></span></a></span>;<br/>
<a NAME="E4:88"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="topgen_5.html#K5" title="TOPGEN_5:func.5">+</a> (<font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c4">y</font>,<font color="Maroon" title="c5">r</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c1">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>,1</span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="topgen_5.html#E29"><span class="lab"><font color="Green" title="E23">Lm1</font></span></a>, <a class="ref" href="borsuk_1.html#T40" title="BORSUK_1:th.40">BORSUK_1:40</a>, <a class="ref" href="funct_2.html#T5" title="FUNCT_2:th.5">FUNCT_2:5</a></span>;<br/>
<a NAME="E5:88"/><span class="kw">then </span><span class="lab"><font color="Green" title="E72">A3</font></span>: 
<span class="p1">(<span class="default"><a href="topgen_5.html#K5" title="TOPGEN_5:func.5">+</a> (<font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c4">y</font>,<font color="Maroon" title="c5">r</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c1">p</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="xxreal_1.html#T1" title="XXREAL_1:th.1">XXREAL_1:1</a></span>;<br/>
<span class="kw">assume </span><a NAME="E6:88"/><span class="lab"><font color="Green" title="E73">A4</font></span>: 
<span class="p1">(<span class="default"><a href="topgen_5.html#K5" title="TOPGEN_5:func.5">+</a> (<font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c4">y</font>,<font color="Maroon" title="c5">r</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c1">p</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c3">a</font>
 ; <a class="txt" 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"><span class="p2">(<span class="default"><span class="p3"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c4">y</font></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <font color="Maroon" title="c1">p</font></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c5">r</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c3">a</font> &amp; <span class="p1">(<span class="default"><a href="topreal9.html#K1" title="TOPREAL9:func.1">Ball</a> (<font color="Maroon" title="c1">p</font>,<span class="p2">(<span class="default"><span class="p3"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><span class="p4">(<span class="default"><span class="p5"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c4">y</font></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <font color="Maroon" title="c1">p</font></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">r</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c3">a</font></span>)</span></span>)</span>)</span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <a href="topgen_5.html#K2" title="TOPGEN_5:func.2">y&gt;=0-plane</a> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><a href="topgen_5.html#K5" title="TOPGEN_5:func.5">+</a> (<font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c4">y</font>,<font color="Maroon" title="c5">r</font>)</span>)</span> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</a> <span class="p1"><a href="rcomp_1.html#K4" title="RCOMP_1:func.4">].</a><span class="default"><font color="Maroon" title="c3">a</font>,1</span><a href="rcomp_1.html#K4" title="RCOMP_1:func.4">.]</a></span> )</span><br/>

<a NAME="E7:88"/><span class="kw">then </span><span class="lab"><font color="Green" title="E74">A5</font></span>: 
<font color="Maroon" title="c3">a</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:88"><span class="lab"><font color="Green" title="E72">A3</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/>
<a NAME="E8:88"/><span class="lab"><font color="Green" title="E75">A6</font></span>: 
<span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c4">y</font></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <font color="Maroon" title="c1">p</font></span>)</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="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <span class="p3"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c4">y</font></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="toprns_1.html#T27" title="TOPRNS_1:th.27">TOPRNS_1:27</a></span>;<br/>
<span class="kw">thus </span><a NAME="E9:88"/>
<span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c4">y</font></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <font color="Maroon" title="c1">p</font></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c5">r</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c3">a</font>
  <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="topreal9.html#K1" title="TOPREAL9:func.1">Ball</a> (<font color="Maroon" title="c1">p</font>,<span class="p2">(<span class="default"><span class="p3"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><span class="p4">(<span class="default"><span class="p5"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c4">y</font></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <font color="Maroon" title="c1">p</font></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">r</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c3">a</font></span>)</span></span>)</span>)</span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <a href="topgen_5.html#K2" title="TOPGEN_5:func.2">y&gt;=0-plane</a> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><a href="topgen_5.html#K5" title="TOPGEN_5:func.5">+</a> (<font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c4">y</font>,<font color="Maroon" title="c5">r</font>)</span>)</span> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</a> <span class="p1"><a href="rcomp_1.html#K4" title="RCOMP_1:func.4">].</a><span class="default"><font color="Maroon" title="c3">a</font>,1</span><a href="rcomp_1.html#K4" title="RCOMP_1:func.4">.]</a></span></span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="88_1"><span class="kw">proof </span></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="88_1_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:88_1_1"/>
( <span class="p1">(<span class="default"><a href="topgen_5.html#K5" title="TOPGEN_5:func.5">+</a> (<font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c4">y</font>,<font color="Maroon" title="c5">r</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c1">p</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1 or <span class="p1">(<span class="default"><a href="topgen_5.html#K5" title="TOPGEN_5:func.5">+</a> (<font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c4">y</font>,<font color="Maroon" title="c5">r</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c1">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:88"><span class="lab"><font color="Green" title="E72">A3</font></span></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="88_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:88_1_1_1"/>
<span class="p1">(<span class="default"><a href="topgen_5.html#K5" title="TOPGEN_5:func.5">+</a> (<font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c4">y</font>,<font color="Maroon" title="c5">r</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c1">p</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1
 ; <a class="txt" 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"><span class="p2">(<span class="default"><span class="p3"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c4">y</font></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <font color="Maroon" title="c1">p</font></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c5">r</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c3">a</font></span><br/><div class="add"><a NAME="E2:88_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c1">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal9.html#K1" title="TOPREAL9:func.1">Ball</a> (<span class="p1"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c4">y</font></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span>,<font color="Maroon" title="c5">r</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:88"><span class="lab"><font color="Green" title="E70">A1</font></span></a>, <a class="txt" href="#E2:88"><span class="lab"><font color="Green" title="E71">A2</font></span></a>, <a class="ref" href="topgen_5.html#D6" target="_self" title="TOPGEN_5:def.6">Def6</a></span>;<br/><a NAME="E3:88_1_1_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="topgen_5.html#K5" title="TOPGEN_5:func.5">+</a> (<font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c4">y</font>,<font color="Maroon" title="c5">r</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c1">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c4">y</font></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <font color="Maroon" title="c1">p</font></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c5">r</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:88"><span class="lab"><font color="Green" title="E70">A1</font></span></a>, <a class="txt" href="#E2:88"><span class="lab"><font color="Green" title="E71">A2</font></span></a>, <a class="ref" href="topgen_5.html#D6" target="_self" title="TOPGEN_5:def.6">Def6</a></span>;<br/><span class="kw">hence </span><a NAME="E4:88_1_1_1"/>
<span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c4">y</font></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <font color="Maroon" title="c1">p</font></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c5">r</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c3">a</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:88"><span class="lab"><font color="Green" title="E73">A4</font></span></a>, <a class="ref" href="xreal_1.html#T79" title="XREAL_1:th.79">XREAL_1:79</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="88_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:88_1_1_2"/><span class="lab"><font color="Green" title="E76">A7</font></span>: 
<span class="p1">(<span class="default"><a href="topgen_5.html#K5" title="TOPGEN_5:func.5">+</a> (<font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c4">y</font>,<font color="Maroon" title="c5">r</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c1">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 ; <a class="txt" 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"><span class="p2">(<span class="default"><span class="p3"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c4">y</font></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <font color="Maroon" title="c1">p</font></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c5">r</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c3">a</font></span><br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="88_1_1_2_1"><span class="kw">now </span></a><div class="add"><a NAME="E1:88_1_1_2_1"/><span class="lab"><font color="Green" title="E77">A8</font></span>: 
<font color="Maroon" title="c5">r</font> <a href="complex1.html#K13" title="COMPLEX1:func.13">/</a> <font color="Maroon" title="c5">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xcmplx_1.html#T60" title="XCMPLX_1:th.60">XCMPLX_1:60</a></span>;<br/><span class="kw">assume </span><a NAME="E2:88_1_1_2_1"/><span class="lab"><font color="Green" title="E78">A9</font></span>: 
<font color="Maroon" title="c1">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal9.html#K1" title="TOPREAL9:func.1">Ball</a> (<span class="p1"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c4">y</font></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span>,<font color="Maroon" title="c5">r</font>)
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><a NAME="E3:88_1_1_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E79">A10</font></span>: 
<span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <span class="p3"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c4">y</font></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c5">r</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="topreal9.html#T7" title="TOPREAL9:th.7">TOPREAL9:7</a></span>;<br/><a NAME="E4:88_1_1_2_1"/>
1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c4">y</font></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <font color="Maroon" title="c1">p</font></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c5">r</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:88_1_1_2_1"><span class="lab"><font color="Green" title="E78">A9</font></span></a>, <a class="txt" href="#E1:88"><span class="lab"><font color="Green" title="E70">A1</font></span></a>, <a class="txt" href="#E2:88"><span class="lab"><font color="Green" title="E71">A2</font></span></a>, <a class="txt" href="#E1:88_1_1_2"><span class="lab"><font color="Green" title="E76">A7</font></span></a>, <a class="ref" href="topgen_5.html#D6" target="_self" title="TOPGEN_5:def.6">Def6</a></span>;<br/><span class="kw">hence </span><a NAME="E5:88_1_1_2_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:88_1_1_2_1"><span class="lab"><font color="Green" title="E79">A10</font></span></a>, <a class="txt" href="#E1:88_1_1_2_1"><span class="lab"><font color="Green" title="E77">A8</font></span></a>, <a class="txt" href="#E8:88"><span class="lab"><font color="Green" title="E75">A6</font></span></a>, <a class="ref" href="xreal_1.html#T74" title="XREAL_1:th.74">XREAL_1:74</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><a NAME="E3:88_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E77">A11</font></span>: 
<span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <span class="p3"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c4">y</font></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c5">r</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="topreal9.html#T7" title="TOPREAL9:th.7">TOPREAL9:7</a></span>;<br/><a NAME="E4:88_1_1_2"/>
<font color="Maroon" title="c5">r</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> 1 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c5">r</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c3">a</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:88"><span class="lab"><font color="Green" title="E74">A5</font></span></a>, <a class="ref" href="xreal_1.html#T68" title="XREAL_1:th.68">XREAL_1:68</a></span>;<br/><span class="kw">hence </span><a NAME="E5:88_1_1_2"/>
<span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c4">y</font></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <font color="Maroon" title="c1">p</font></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c5">r</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c3">a</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:88_1_1_2"><span class="lab"><font color="Green" title="E77">A11</font></span></a>, <a class="txt" href="#E8:88"><span class="lab"><font color="Green" title="E75">A6</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

</div><span class="kw">end;</span></div>

<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c7">r1</font> = <span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c4">y</font></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <font color="Maroon" title="c1">p</font></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">r</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c3">a</font></span>)</span> as   <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>   <a href="xxreal_0.html#V2" title="XXREAL_0:attr.2">positive</a>   <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>  <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T50" title="XREAL_1:th.50">XREAL_1:50</a></span>;<br/>
<span class="kw">let </span><font color="Maroon" title="c8">u</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c8">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="topreal9.html#K1" title="TOPREAL9:func.1">Ball</a> (<font color="Maroon" title="c1">p</font>,<span class="p2">(<span class="default"><span class="p3"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><span class="p4">(<span class="default"><span class="p5"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c4">y</font></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <font color="Maroon" title="c1">p</font></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">r</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c3">a</font></span>)</span></span>)</span>)</span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <a href="topgen_5.html#K2" title="TOPGEN_5:func.2">y&gt;=0-plane</a> or <font color="Maroon" title="c8">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="topgen_5.html#K5" title="TOPGEN_5:func.5">+</a> (<font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c4">y</font>,<font color="Maroon" title="c5">r</font>)</span>)</span> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</a> <span class="p1"><a href="rcomp_1.html#K4" title="RCOMP_1:func.4">].</a><span class="default"><font color="Maroon" title="c3">a</font>,1</span><a href="rcomp_1.html#K4" title="RCOMP_1:func.4">.]</a></span> )</span><br/>

<span class="kw">assume </span><a NAME="E11:88"/><span class="lab"><font color="Green" title="E76">A12</font></span>: 
<font color="Maroon" title="c8">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="topreal9.html#K1" title="TOPREAL9:func.1">Ball</a> (<font color="Maroon" title="c1">p</font>,<span class="p2">(<span class="default"><span class="p3"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><span class="p4">(<span class="default"><span class="p5"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c4">y</font></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <font color="Maroon" title="c1">p</font></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">r</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c3">a</font></span>)</span></span>)</span>)</span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <a href="topgen_5.html#K2" title="TOPGEN_5:func.2">y&gt;=0-plane</a>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c8">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="topgen_5.html#K5" title="TOPGEN_5:func.5">+</a> (<font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c4">y</font>,<font color="Maroon" title="c5">r</font>)</span>)</span> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</a> <span class="p1"><a href="rcomp_1.html#K4" title="RCOMP_1:func.4">].</a><span class="default"><font color="Maroon" title="c3">a</font>,1</span><a href="rcomp_1.html#K4" title="RCOMP_1:func.4">.]</a></span></span><br/>

<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c9">z</font> = <font color="Maroon" title="c8">u</font> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <a href="topgen_5.html#K3" title="TOPGEN_5:func.3">Niemytzki-plane</a> <span class="kw">by</span> <span class="lab"><a class="txt" href="topgen_5.html#E29"><span class="lab"><font color="Green" title="E23">Lm1</font></span></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c10">q</font> = <font color="Maroon" title="c9">z</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:88"><span class="lab"><font color="Green" title="E76">A12</font></span></a></span>;<br/>
<a NAME="E14:88"/>
<font color="Maroon" title="c8">u</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="c1">p</font>,<span class="p1">(<span class="default"><span class="p2"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><span class="p3">(<span class="default"><span class="p4"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c4">y</font></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <font color="Maroon" title="c1">p</font></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">r</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c3">a</font></span>)</span></span>)</span>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:88"><span class="lab"><font color="Green" title="E76">A12</font></span></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>;<br/>
<a NAME="E15:88"/><span class="kw">then </span>
<span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c10">q</font> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <font color="Maroon" title="c1">p</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="c7">r1</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="topreal9.html#T7" title="TOPREAL9:th.7">TOPREAL9:7</a></span>;<br/>
<a NAME="E16:88"/><span class="kw">then </span><span class="lab"><font color="Green" title="E77">A13</font></span>: 
<span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c4">y</font></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <font color="Maroon" title="c10">q</font></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c10">q</font> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <font color="Maroon" title="c1">p</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> <span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c4">y</font></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <font color="Maroon" title="c10">q</font></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c7">r1</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T6" title="XREAL_1:th.6">XREAL_1:6</a></span>;<br/>
<a NAME="E17:88"/><span class="lab"><font color="Green" title="E78">A14</font></span>: 
<font color="Maroon" title="c10">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c10">q</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c10">q</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="euclid.html#T53" title="EUCLID:th.53">EUCLID:53</a></span>;<br/>
<a NAME="E18:88"/><span class="kw">then </span><span class="lab"><font color="Green" title="E79">A15</font></span>: 
<font color="Maroon" title="c10">q</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="topgen_5.html#E29"><span class="lab"><font color="Green" title="E23">Lm1</font></span></a>, <a class="ref" href="topgen_5.html#T18" target="_self" title="TOPGEN_5:th.18">Th22</a></span>;<br/>
<a NAME="E19:88"/>
<span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c4">y</font></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <font color="Maroon" title="c1">p</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> <span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c4">y</font></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <font color="Maroon" title="c10">q</font></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c10">q</font> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <font color="Maroon" title="c1">p</font></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="toprns_1.html#T34" title="TOPRNS_1:th.34">TOPRNS_1:34</a></span>;<br/>
<a NAME="E20:88"/><span class="kw">then </span>
<span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c4">y</font></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <font color="Maroon" title="c1">p</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> <span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c4">y</font></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <font color="Maroon" title="c10">q</font></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c7">r1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:88"><span class="lab"><font color="Green" title="E77">A13</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/>
<a NAME="E21:88"/><span class="kw">then </span><span class="lab"><font color="Green" title="E80">A16</font></span>: 
<span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c4">y</font></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <font color="Maroon" title="c1">p</font></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c7">r1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><span class="p2"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><span class="p3">(<span class="default"><span class="p4"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c4">y</font></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <font color="Maroon" title="c10">q</font></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c7">r1</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c7">r1</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T9" title="XREAL_1:th.9">XREAL_1:9</a></span>;<br/>
<div><span class="lab"><font color="Green" title="E81">A17</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="88_2"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:88_2"/>
<font color="Maroon" title="c10">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal9.html#K1" title="TOPREAL9:func.1">Ball</a> (<span class="p1"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c4">y</font></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span>,<font color="Maroon" title="c5">r</font>)
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="topgen_5.html#K5" title="TOPGEN_5:func.5">+</a> (<font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c4">y</font>,<font color="Maroon" title="c5">r</font>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">z</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c3">a</font></span><br/><a NAME="E2:88_2"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="topgen_5.html#K5" title="TOPGEN_5:func.5">+</a> (<font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c4">y</font>,<font color="Maroon" title="c5">r</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c4">y</font></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <font color="Maroon" title="c10">q</font></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c5">r</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:88"><span class="lab"><font color="Green" title="E78">A14</font></span></a>, <a class="txt" href="#E18:88"><span class="lab"><font color="Green" title="E79">A15</font></span></a>, <a class="ref" href="topgen_5.html#D6" target="_self" title="TOPGEN_5:def.6">Def6</a></span>;<br/><span class="kw">hence </span><a NAME="E3:88_2"/>
<span class="p1">(<span class="default"><a href="topgen_5.html#K5" title="TOPGEN_5:func.5">+</a> (<font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c4">y</font>,<font color="Maroon" title="c5">r</font>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">z</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c3">a</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E21:88"><span class="lab"><font color="Green" title="E80">A16</font></span></a>, <a class="ref" href="xreal_1.html#T81" title="XREAL_1:th.81">XREAL_1:81</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<a NAME="E23:88"/>
<span class="p1">(<span class="default"><a href="topgen_5.html#K5" title="TOPGEN_5:func.5">+</a> (<font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c4">y</font>,<font color="Maroon" title="c5">r</font>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>,1</span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="borsuk_1.html#T40" title="BORSUK_1:th.40">BORSUK_1:40</a></span>;<br/>
<a NAME="E24:88"/><span class="kw">then </span><span class="lab"><font color="Green" title="E82">A18</font></span>: 
<span class="p1">(<span class="default"><a href="topgen_5.html#K5" title="TOPGEN_5:func.5">+</a> (<font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c4">y</font>,<font color="Maroon" title="c5">r</font>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">z</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="xxreal_1.html#T1" title="XXREAL_1:th.1">XXREAL_1:1</a></span>;<br/>
<a NAME="E25:88"/>
(  not <font color="Maroon" title="c10">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal9.html#K1" title="TOPREAL9:func.1">Ball</a> (<span class="p1"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c4">y</font></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span>,<font color="Maroon" title="c5">r</font>) implies <span class="p1">(<span class="default"><a href="topgen_5.html#K5" title="TOPGEN_5:func.5">+</a> (<font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c4">y</font>,<font color="Maroon" title="c5">r</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:88"><span class="lab"><font color="Green" title="E79">A15</font></span></a>, <a class="txt" href="#E17:88"><span class="lab"><font color="Green" title="E78">A14</font></span></a>, <a class="ref" href="topgen_5.html#D6" target="_self" title="TOPGEN_5:def.6">Def6</a></span>;<br/>
<a NAME="E26:88"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="topgen_5.html#K5" title="TOPGEN_5:func.5">+</a> (<font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c4">y</font>,<font color="Maroon" title="c5">r</font>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="rcomp_1.html#K4" title="RCOMP_1:func.4">].</a><span class="default"><font color="Maroon" title="c3">a</font>,1</span><a href="rcomp_1.html#K4" title="RCOMP_1:func.4">.]</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E24:88"><span class="lab"><font color="Green" title="E82">A18</font></span></a>, <a class="txt" href="#E7:88"><span class="lab"><font color="Green" title="E74">A5</font></span></a>, <a class="txt" href="#E22:88"><span class="lab"><font color="Green" title="E81">A17</font></span></a>, <a class="ref" href="xxreal_1.html#T2" title="XXREAL_1:th.2">XXREAL_1:2</a></span>;<br/>
<span class="kw">hence </span><a NAME="E27:88"/>
<font color="Maroon" title="c8">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="topgen_5.html#K5" title="TOPGEN_5:func.5">+</a> (<font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c4">y</font>,<font color="Maroon" title="c5">r</font>)</span>)</span> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</a> <span class="p1"><a href="rcomp_1.html#K4" title="RCOMP_1:func.4">].</a><span class="default"><font color="Maroon" title="c3">a</font>,1</span><a href="rcomp_1.html#K4" title="RCOMP_1:func.4">.]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#T38" title="FUNCT_2:th.38">FUNCT_2:38</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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