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<div class="add">

<b>reconsider </b><font color="Maroon">c<sub>4</sub></font> = <font color="Maroon">c<sub>2</sub></font> as   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K14">Euclid</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <b>by </b><i><a class="ref" href="topreal3.html#T13">TOPREAL3:13</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="56_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:56_1_1_1"/>
( <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> 0 or <font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="56_1_1_1_1"><b>suppose </b></a><a NAME="E1:56_1_1_1_1"/>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> 0
 ;<br/><div class="add"><a NAME="E2:56_1_1_1_1"/><b>then </b>
 <a href="metric_1.html#K11">Sphere</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>4</sub></font></span><a href="domain_1.html#K6">}</a></span>
 <b>by </b><i><a class="ref" href="topreal6.html#T62">TOPREAL6:62</a></i>;<br/><b>hence </b><a NAME="E3:56_1_1_1_1"/>
not  <a href="topreal9.html#K3" target="_self">Sphere</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font> is <a href="xboole_0.html#V1">empty</a>
 <b>by </b><i><a class="ref" href="topreal9.html#T15" target="_self">Th15</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="56_1_1_1_2"><b>suppose </b></a><a NAME="E1:56_1_1_1_2"/>
<font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 ;<br/><div class="add"><b>then reconsider </b><font color="Maroon">c<sub>5</sub></font> = <font color="Maroon">c<sub>3</sub></font> as  <a href="xxreal_0.html#V2">positive</a> <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a>  ;<br/><b>consider </b><font color="Maroon">c<sub>6</sub></font> being   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> <font color="Maroon">c<sub>1</sub></font></span>)</span><b> such that </b><br/><a NAME="E4:56_1_1_1_2"/><i><font color="Green">E33</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a>  <a href="topreal9.html#K1" target="_self">Ball</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="ref" href="subset_1.html#T10">SUBSET_1:10</a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>7</sub></font> = <font color="Maroon">c<sub>6</sub></font>, <font color="Maroon">c<sub>8</sub></font> = <font color="Maroon">c<sub>6</sub></font> <a href="euclid.html#K17">+</a> <span class="p1">(<span class="default"><a href="jordan2c.html#K5">1.REAL</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>, <font color="Maroon">c<sub>9</sub></font> = <font color="Maroon">c<sub>2</sub></font> as    <a href="finseq_2.html#M2">Element</a> of  <a href="euclid.html#K1">REAL</a> <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="ref" href="euclid.html#T25">EUCLID:25</a></i>;<br/><b>set </b><font color="Maroon">c<sub>10</sub></font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <span class="p3">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <span class="p4"><a href="euclid_2.html#K2">|(</a><span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="euclid.html#K17">+</a> <span class="p1">(<span class="default"><a href="jordan2c.html#K5">1.REAL</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="euclid.html#K20">-</a> <font color="Maroon">c<sub>6</sub></font></span>)</span>,<span class="p5">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="euclid.html#K20">-</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span><a href="euclid_2.html#K2">)|</a></span></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><a href="square_1.html#K6">sqrt</a> <span class="p3">(<span class="default"><a href="quin_1.html#K1">delta</a> <span class="p4">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <span class="p5">(<span class="default"><a href="euclid.html#K11">sqr</a> <span class="p0">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span></span>)</span>,<span class="p4">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <span class="p5"><a href="euclid_2.html#K2">|(</a><span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="euclid.html#K17">+</a> <span class="p2">(<span class="default"><a href="jordan2c.html#K5">1.REAL</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="euclid.html#K20">-</a> <font color="Maroon">c<sub>6</sub></font></span>)</span>,<span class="p0">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="euclid.html#K20">-</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span><a href="euclid_2.html#K2">)|</a></span></span>)</span>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <span class="p0">(<span class="default"><a href="euclid.html#K11">sqr</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>9</sub></font></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p5">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="square_1.html#K3">^2</a> </span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <span class="p3">(<span class="default"><a href="euclid.html#K11">sqr</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span></span>)</span></span>)</span>;<br/><a NAME="E6:56_1_1_1_2"/>
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>6</sub></font> <a href="euclid.html#K17">+</a> <span class="p1">(<span class="default"><a href="jordan2c.html#K5">1.REAL</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="topreal9.html#T3" target="_self">Th3</a></i>;<br/><a NAME="E7:56_1_1_1_2"/><b>then </b>
ex <font color="Olive">b<sub>1</sub></font> being  <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> st <br/>( <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Olive">b<sub>1</sub></font></span><a href="domain_1.html#K6">}</a></span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="topreal9.html#K4" target="_self">halfline</a> <font color="Maroon">c<sub>6</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="euclid.html#K17">+</a> <span class="p3">(<span class="default"><a href="jordan2c.html#K5">1.REAL</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="topreal9.html#K3" target="_self">Sphere</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>5</sub></font></span>)</span> &amp; <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="xcmplx_0.html#K6">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <span class="p0">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <span class="p1"><a href="euclid_2.html#K2">|(</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="euclid.html#K17">+</a> <span class="p4">(<span class="default"><a href="jordan2c.html#K5">1.REAL</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="euclid.html#K20">-</a> <font color="Maroon">c<sub>6</sub></font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="euclid.html#K20">-</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span><a href="euclid_2.html#K2">)|</a></span></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p5">(<span class="default"><a href="square_1.html#K6">sqrt</a> <span class="p0">(<span class="default"><a href="quin_1.html#K1">delta</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <span class="p2">(<span class="default"><a href="euclid.html#K11">sqr</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span></span>)</span>,<span class="p1">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <span class="p2"><a href="euclid_2.html#K2">|(</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="euclid.html#K17">+</a> <span class="p5">(<span class="default"><a href="jordan2c.html#K5">1.REAL</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="euclid.html#K20">-</a> <font color="Maroon">c<sub>6</sub></font></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="euclid.html#K20">-</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span><a href="euclid_2.html#K2">)|</a></span></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <span class="p3">(<span class="default"><a href="euclid.html#K11">sqr</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>9</sub></font></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="square_1.html#K3">^2</a> </span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p4">(<span class="default">2 <a href="real_1.html#K4">*</a> <span class="p5">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <span class="p0">(<span class="default"><a href="euclid.html#K11">sqr</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="euclid.html#K17">+</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="xcmplx_0.html#K4">-</a> <span class="p5">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <span class="p0"><a href="euclid_2.html#K2">|(</a><span class="default"><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="euclid.html#K17">+</a> <span class="p3">(<span class="default"><a href="jordan2c.html#K5">1.REAL</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="euclid.html#K20">-</a> <font color="Maroon">c<sub>6</sub></font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="euclid.html#K20">-</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span><a href="euclid_2.html#K2">)|</a></span></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p4">(<span class="default"><a href="square_1.html#K6">sqrt</a> <span class="p5">(<span class="default"><a href="quin_1.html#K1">delta</a> <span class="p0">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <span class="p1">(<span class="default"><a href="euclid.html#K11">sqr</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span></span>)</span>,<span class="p0">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <span class="p1"><a href="euclid_2.html#K2">|(</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="euclid.html#K17">+</a> <span class="p4">(<span class="default"><a href="jordan2c.html#K5">1.REAL</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="euclid.html#K20">-</a> <font color="Maroon">c<sub>6</sub></font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="euclid.html#K20">-</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span><a href="euclid_2.html#K2">)|</a></span></span>)</span>,<span class="p0">(<span class="default"><span class="p1">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <span class="p2">(<span class="default"><a href="euclid.html#K11">sqr</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>9</sub></font></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="square_1.html#K3">^2</a> </span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p3">(<span class="default">2 <a href="real_1.html#K4">*</a> <span class="p4">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <span class="p5">(<span class="default"><a href="euclid.html#K11">sqr</a> <span class="p0">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K18">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="euclid.html#K17">+</a> <span class="p3">(<span class="default"><a href="jordan2c.html#K5">1.REAL</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span> )
 <b>by </b><i><a class="txt" href="topreal9.html#E4:56_1_1_1_2"><i><font color="Green">E33</font></i></a>, <a class="ref" href="topreal9.html#T37" target="_self">Th37</a></i>;<br/><b>hence </b><a NAME="E8:56_1_1_1_2"/>
not  <a href="topreal9.html#K3" target="_self">Sphere</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font> is <a href="xboole_0.html#V1">empty</a>
 ;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div>
