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<span class="kw">let </span><font color="Maroon" title="c22">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="c22">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="jgraph_6.html#K2" title="JGRAPH_6:func.2">closed_inside_of_rectangle</a> (<span class="p1">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,1,<span class="p1">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 3</span>)</span>,3) implies  <a href="jgraph_6.html#K2" title="JGRAPH_6:func.2">closed_inside_of_rectangle</a> (<span class="p1">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,1,<span class="p1">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 3</span>)</span>,3) <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="topreal9.html#K1" title="TOPREAL9:func.1">Ball</a> (<font color="Maroon" title="c22">p</font>,10) )</span><br/>

<span class="kw">assume </span><a NAME="E1:144"/>
<font color="Maroon" title="c22">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="jgraph_6.html#K2" title="JGRAPH_6:func.2">closed_inside_of_rectangle</a> (<span class="p1">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,1,<span class="p1">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 3</span>)</span>,3)
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="jgraph_6.html#K2" title="JGRAPH_6:func.2">closed_inside_of_rectangle</a> (<span class="p1">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,1,<span class="p1">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 3</span>)</span>,3) <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="topreal9.html#K1" title="TOPREAL9:func.1">Ball</a> (<font color="Maroon" title="c22">p</font>,10)</span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c23">p1</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> 2</span>)</span><span class="kw"> such that </span><br/><a NAME="E3:144"/><span class="lab"><font color="Green" title="E176">A1</font></span>: 
<font color="Maroon" title="c23">p1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">p</font>
 <span class="kw">and </span><br/><a NAME="E4:144"/><span class="lab"><font color="Green" title="E177">A2</font></span>: 
 <a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c23">p1</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> 
 <span class="kw">and </span><br/><a NAME="E5:144"/><span class="lab"><font color="Green" title="E178">A3</font></span>: 
<font color="Maroon" title="c23">p1</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1
 <span class="kw">and </span><br/><a NAME="E6:144"/><span class="lab"><font color="Green" title="E179">A4</font></span>: 
 <a href="real_1.html#K1" title="REAL_1:func.1">-</a> 3 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c23">p1</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> 
 <span class="kw">and </span><br/><a NAME="E7:144"/><span class="lab"><font color="Green" title="E180">A5</font></span>: 
<font color="Maroon" title="c23">p1</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 3
 ;<br/>
<span class="kw">let </span><font color="Maroon" title="c24">x</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="c24">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="jgraph_6.html#K2" title="JGRAPH_6:func.2">closed_inside_of_rectangle</a> (<span class="p1">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,1,<span class="p1">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 3</span>)</span>,3) or <font color="Maroon" title="c24">x</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="c22">p</font>,10) )</span><br/>

<span class="kw">assume </span><a NAME="E8:144"/><span class="lab"><font color="Green" title="E181">A6</font></span>: 
<font color="Maroon" title="c24">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="jgraph_6.html#K2" title="JGRAPH_6:func.2">closed_inside_of_rectangle</a> (<span class="p1">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,1,<span class="p1">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 3</span>)</span>,3)
 ; <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="c24">x</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="c22">p</font>,10)</span><br/>

<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c25">x</font> = <font color="Maroon" title="c24">x</font> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> ;<br/>
<span class="kw">consider </span><font color="Maroon" title="c26">p2</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> 2</span>)</span><span class="kw"> such that </span><br/><a NAME="E11:144"/><span class="lab"><font color="Green" title="E182">A7</font></span>: 
<font color="Maroon" title="c26">p2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c25">x</font>
 <span class="kw">and </span><br/><a NAME="E12:144"/><span class="lab"><font color="Green" title="E183">A8</font></span>: 
 <a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c26">p2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> 
 <span class="kw">and </span><br/><a NAME="E13:144"/><span class="lab"><font color="Green" title="E184">A9</font></span>: 
<font color="Maroon" title="c26">p2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1
 <span class="kw">and </span><br/><a NAME="E14:144"/><span class="lab"><font color="Green" title="E185">A10</font></span>: 
 <a href="real_1.html#K1" title="REAL_1:func.1">-</a> 3 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c26">p2</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> 
 <span class="kw">and </span><br/><a NAME="E15:144"/><span class="lab"><font color="Green" title="E186">A11</font></span>: 
<font color="Maroon" title="c26">p2</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 3
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:144"><span class="lab"><font color="Green" title="E181">A6</font></span></a></span>;<br/>
<a NAME="E16:144"/><span class="lab"><font color="Green" title="E187">A12</font></span>: 
 ex <font color="Olive" title="b1">s</font>, <font color="Olive" title="b2">t</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#K14" title="EUCLID:func.14">Euclid</a> 2</span>)</span> st <br/>( <font color="Olive" title="b1">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">p1</font> &amp; <font color="Olive" title="b2">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c26">p2</font> &amp;  <a href="topreal6.html#K1" title="TOPREAL6:func.1">dist</a> (<font color="Maroon" title="c23">p1</font>,<font color="Maroon" title="c26">p2</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> (<font color="Olive" title="b1">s</font>,<font color="Olive" title="b2">t</font>) )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="topreal6.html#D1" title="TOPREAL6:def.1">TOPREAL6:def 1</a></span>;<br/>
<a NAME="E17:144"/>
 <a href="topreal6.html#K1" title="TOPREAL6:func.1">dist</a> (<font color="Maroon" title="c23">p1</font>,<font color="Maroon" title="c26">p2</font>) <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default">3 <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 3</span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:144"><span class="lab"><font color="Green" title="E177">A2</font></span></a>, <a class="txt" href="#E5:144"><span class="lab"><font color="Green" title="E178">A3</font></span></a>, <a class="txt" href="#E6:144"><span class="lab"><font color="Green" title="E179">A4</font></span></a>, <a class="txt" href="#E7:144"><span class="lab"><font color="Green" title="E180">A5</font></span></a>, <a class="txt" href="#E12:144"><span class="lab"><font color="Green" title="E183">A8</font></span></a>, <a class="txt" href="#E13:144"><span class="lab"><font color="Green" title="E184">A9</font></span></a>, <a class="txt" href="#E14:144"><span class="lab"><font color="Green" title="E185">A10</font></span></a>, <a class="txt" href="#E15:144"><span class="lab"><font color="Green" title="E186">A11</font></span></a>, <a class="ref" href="topreal6.html#T95" title="TOPREAL6:th.95">TOPREAL6:95</a></span>;<br/>
<a NAME="E18:144"/><span class="kw">then </span>
 <a href="topreal6.html#K1" title="TOPREAL6:func.1">dist</a> (<font color="Maroon" title="c23">p1</font>,<font color="Maroon" title="c26">p2</font>) <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 10
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/>
<a NAME="E19:144"/><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="c25">x</font> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <font color="Maroon" title="c22">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> 10
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:144"><span class="lab"><font color="Green" title="E176">A1</font></span></a>, <a class="txt" href="#E11:144"><span class="lab"><font color="Green" title="E182">A7</font></span></a>, <a class="txt" href="#E16:144"><span class="lab"><font color="Green" title="E187">A12</font></span></a>, <a class="ref" href="sppol_1.html#T39" title="SPPOL_1:th.39">SPPOL_1:39</a></span>;<br/>
<span class="kw">hence </span><a NAME="E20:144"/>
<font color="Maroon" title="c24">x</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="c22">p</font>,10)
 <span class="kw">by</span> <span class="lab"><a class="ref" href="topreal9.html#T7" title="TOPREAL9:th.7">TOPREAL9:7</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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