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<b>let </b><font color="Maroon" title="c1">P</font>, <font color="Maroon" title="c2">P1</font> be  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="compts_1.html#V6" title="COMPTS_1:attr.6">compact</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>;<br/>

<b>assume </b><a NAME="E1:16"/><i><font color="Green" title="E14">A1</font></i>: 
( <font color="Maroon" title="c1">P</font> <a href="topreal2.html#NR1" title="TOPREAL2:NR.1">is_simple_closed_curve</a>  &amp; <font color="Maroon" title="c2">P1</font> <a href="topreal1.html#R1" title="TOPREAL1:pred.1">is_an_arc_of</a>  <a href="pscomp_1.html#K25" title="PSCOMP_1:func.25">W-min</a> <font color="Maroon" title="c1">P</font>, <a href="pscomp_1.html#K29" title="PSCOMP_1:func.29">E-max</a> <font color="Maroon" title="c1">P</font> &amp; <font color="Maroon" title="c2">P1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">P</font> )
 ;<br/>

<a NAME="E2:16"/><b>then </b><i><font color="Green" title="E15">A2</font></i>: 
(  <a href="jordan6.html#K8" title="JORDAN6:func.8">Upper_Arc</a> <font color="Maroon" title="c1">P</font> <a href="topreal1.html#R1" title="TOPREAL1:pred.1">is_an_arc_of</a>  <a href="pscomp_1.html#K25" title="PSCOMP_1:func.25">W-min</a> <font color="Maroon" title="c1">P</font>, <a href="pscomp_1.html#K29" title="PSCOMP_1:func.29">E-max</a> <font color="Maroon" title="c1">P</font> &amp;  ex <font color="Olive" title="b1">P2</font> being non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> st <br/>( <font color="Olive" title="b1">P2</font> <a href="topreal1.html#R1" title="TOPREAL1:pred.1">is_an_arc_of</a>  <a href="pscomp_1.html#K29" title="PSCOMP_1:func.29">E-max</a> <font color="Maroon" title="c1">P</font>, <a href="pscomp_1.html#K25" title="PSCOMP_1:func.25">W-min</a> <font color="Maroon" title="c1">P</font> &amp; <span class="p1">(<span class="default"><a href="jordan6.html#K8" title="JORDAN6:func.8">Upper_Arc</a> <font color="Maroon" title="c1">P</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Olive" title="b1">P2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K25" title="PSCOMP_1:func.25">W-min</a> <font color="Maroon" title="c1">P</font></span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K29" title="PSCOMP_1:func.29">E-max</a> <font color="Maroon" title="c1">P</font></span>)</span></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> &amp; <span class="p1">(<span class="default"><a href="jordan6.html#K8" title="JORDAN6:func.8">Upper_Arc</a> <font color="Maroon" title="c1">P</font></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Olive" title="b1">P2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">P</font> &amp; <span class="p1">(<span class="default"><a href="jordan5c.html#K1" title="JORDAN5C:func.1">First_Point</a> <span class="p2">(<span class="default"><a href="jordan6.html#K8" title="JORDAN6:func.8">Upper_Arc</a> <font color="Maroon" title="c1">P</font></span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K25" title="PSCOMP_1:func.25">W-min</a> <font color="Maroon" title="c1">P</font></span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K29" title="PSCOMP_1:func.29">E-max</a> <font color="Maroon" title="c1">P</font></span>)</span>,<span class="p2">(<span class="default"><a href="jordan6.html#K6" title="JORDAN6:func.6">Vertical_Line</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="pscomp_1.html#K13" title="PSCOMP_1:func.13">W-bound</a> <font color="Maroon" title="c1">P</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p5">(<span class="default"><a href="pscomp_1.html#K15" title="PSCOMP_1:func.15">E-bound</a> <font color="Maroon" title="c1">P</font></span>)</span></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <span class="p1">(<span class="default"><a href="jordan5c.html#K2" title="JORDAN5C:func.2">Last_Point</a> <font color="Olive" title="b1">P2</font>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K29" title="PSCOMP_1:func.29">E-max</a> <font color="Maroon" title="c1">P</font></span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K25" title="PSCOMP_1:func.25">W-min</a> <font color="Maroon" title="c1">P</font></span>)</span>,<span class="p2">(<span class="default"><a href="jordan6.html#K6" title="JORDAN6:func.6">Vertical_Line</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="pscomp_1.html#K13" title="PSCOMP_1:func.13">W-bound</a> <font color="Maroon" title="c1">P</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p5">(<span class="default"><a href="pscomp_1.html#K15" title="PSCOMP_1:func.15">E-bound</a> <font color="Maroon" title="c1">P</font></span>)</span></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  ) )
 
<b>by </b><i><a class="ref" href="jordan6.html#D8" title="JORDAN6:def.8">JORDAN6:def 8</a></i>;<br/>
<a NAME="E3:16"/><i><font color="Green" title="E16">A3</font></i>: 
(  <a href="jordan6.html#K9" title="JORDAN6:func.9">Lower_Arc</a> <font color="Maroon" title="c1">P</font> <a href="topreal1.html#R1" title="TOPREAL1:pred.1">is_an_arc_of</a>  <a href="pscomp_1.html#K29" title="PSCOMP_1:func.29">E-max</a> <font color="Maroon" title="c1">P</font>, <a href="pscomp_1.html#K25" title="PSCOMP_1:func.25">W-min</a> <font color="Maroon" title="c1">P</font> &amp; <span class="p1">(<span class="default"><a href="jordan6.html#K8" title="JORDAN6:func.8">Upper_Arc</a> <font color="Maroon" title="c1">P</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="jordan6.html#K9" title="JORDAN6:func.9">Lower_Arc</a> <font color="Maroon" title="c1">P</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K25" title="PSCOMP_1:func.25">W-min</a> <font color="Maroon" title="c1">P</font></span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K29" title="PSCOMP_1:func.29">E-max</a> <font color="Maroon" title="c1">P</font></span>)</span></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> &amp; <span class="p1">(<span class="default"><a href="jordan6.html#K8" title="JORDAN6:func.8">Upper_Arc</a> <font color="Maroon" title="c1">P</font></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><a href="jordan6.html#K9" title="JORDAN6:func.9">Lower_Arc</a> <font color="Maroon" title="c1">P</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">P</font> &amp; <span class="p1">(<span class="default"><a href="jordan5c.html#K1" title="JORDAN5C:func.1">First_Point</a> <span class="p2">(<span class="default"><a href="jordan6.html#K8" title="JORDAN6:func.8">Upper_Arc</a> <font color="Maroon" title="c1">P</font></span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K25" title="PSCOMP_1:func.25">W-min</a> <font color="Maroon" title="c1">P</font></span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K29" title="PSCOMP_1:func.29">E-max</a> <font color="Maroon" title="c1">P</font></span>)</span>,<span class="p2">(<span class="default"><a href="jordan6.html#K6" title="JORDAN6:func.6">Vertical_Line</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="pscomp_1.html#K13" title="PSCOMP_1:func.13">W-bound</a> <font color="Maroon" title="c1">P</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p5">(<span class="default"><a href="pscomp_1.html#K15" title="PSCOMP_1:func.15">E-bound</a> <font color="Maroon" title="c1">P</font></span>)</span></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <span class="p1">(<span class="default"><a href="jordan5c.html#K2" title="JORDAN5C:func.2">Last_Point</a> <span class="p2">(<span class="default"><a href="jordan6.html#K9" title="JORDAN6:func.9">Lower_Arc</a> <font color="Maroon" title="c1">P</font></span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K29" title="PSCOMP_1:func.29">E-max</a> <font color="Maroon" title="c1">P</font></span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K25" title="PSCOMP_1:func.25">W-min</a> <font color="Maroon" title="c1">P</font></span>)</span>,<span class="p2">(<span class="default"><a href="jordan6.html#K6" title="JORDAN6:func.6">Vertical_Line</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="pscomp_1.html#K13" title="PSCOMP_1:func.13">W-bound</a> <font color="Maroon" title="c1">P</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p5">(<span class="default"><a href="pscomp_1.html#K15" title="PSCOMP_1:func.15">E-bound</a> <font color="Maroon" title="c1">P</font></span>)</span></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  )
 
<b>by </b><i><a class="txt" href="topmetr3.html#E1:16"><i><font color="Green" title="E14">A1</font></i></a>, <a class="ref" href="jordan6.html#D9" title="JORDAN6:def.9">JORDAN6:def 9</a></i>;<br/>
<a NAME="E4:16"/><b>then </b><i><font color="Green" title="E17">A4</font></i>: 
 <a href="pscomp_1.html#K29" title="PSCOMP_1:func.29">E-max</a> <font color="Maroon" title="c1">P</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="jordan6.html#K8" title="JORDAN6:func.8">Upper_Arc</a> <font color="Maroon" title="c1">P</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="jordan6.html#K9" title="JORDAN6:func.9">Lower_Arc</a> <font color="Maroon" title="c1">P</font></span>)</span>
 
<b>by </b><i><a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a></i>;<br/>
<a NAME="E5:16"/><i><font color="Green" title="E18">A5</font></i>: 
 <a href="pscomp_1.html#K25" title="PSCOMP_1:func.25">W-min</a> <font color="Maroon" title="c1">P</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="jordan6.html#K8" title="JORDAN6:func.8">Upper_Arc</a> <font color="Maroon" title="c1">P</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="jordan6.html#K9" title="JORDAN6:func.9">Lower_Arc</a> <font color="Maroon" title="c1">P</font></span>)</span>
 
<b>by </b><i><a class="txt" href="topmetr3.html#E3:16"><i><font color="Green" title="E16">A3</font></i></a>, <a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c3">f</font> being   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> ,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c2">P1</font></span>)</span><b> such that </b><br/><a NAME="E7:16"/><i><font color="Green" title="E19">A6</font></i>: 
( <font color="Maroon" title="c3">f</font> <a href="tops_2.html#NR1" title="TOPS_2:NR.1">is_homeomorphism</a>  &amp; <font color="Maroon" title="c3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pscomp_1.html#K25" title="PSCOMP_1:func.25">W-min</a> <font color="Maroon" title="c1">P</font> &amp; <font color="Maroon" title="c3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pscomp_1.html#K29" title="PSCOMP_1:func.29">E-max</a> <font color="Maroon" title="c1">P</font> )
 <b>by </b><i><a class="txt" href="topmetr3.html#E1:16"><i><font color="Green" title="E14">A1</font></i></a>, <a class="ref" href="topreal1.html#D2" title="TOPREAL1:def.2">TOPREAL1:def 2</a></i>;<br/>
<i><font color="Green" title="E20">A7</font></i>: the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c1">P</font></span>)</span> = 
 <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c1">P</font></span>)</span>
<b>by </b><i><a class="ref" href="pre_topc.html#T12" title="PRE_TOPC:th.12">PRE_TOPC:12</a></i>
<br/>.= 
<font color="Maroon" title="c1">P</font>
<b>by </b><i><a class="ref" href="pre_topc.html#D10" title="PRE_TOPC:def.10">PRE_TOPC:def 10</a></i>
;<br/>
<a NAME="E9:16"/><i><font color="Green" title="E21">A8</font></i>: 
<font color="Maroon" title="c3">f</font> is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>
 
<b>by </b><i><a class="txt" href="topmetr3.html#E7:16"><i><font color="Green" title="E19">A6</font></i></a>, <a class="ref" href="tops_2.html#D5" title="TOPS_2:def.5">TOPS_2:def 5</a></i>;<br/>
<a NAME="E10:16"/><i><font color="Green" title="E22">A9</font></i>: 
<font color="Maroon" title="c3">f</font> is <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a>
 
<b>by </b><i><a class="txt" href="topmetr3.html#E7:16"><i><font color="Green" title="E19">A6</font></i></a>, <a class="ref" href="tops_2.html#D5" title="TOPS_2:def.5">TOPS_2:def 5</a></i>;<br/>
<a NAME="E11:16"/>
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c3">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> 
 
<b>by </b><i><a class="txt" href="topmetr3.html#E7:16"><i><font color="Green" title="E19">A6</font></i></a>, <a class="ref" href="tops_2.html#D5" title="TOPS_2:def.5">TOPS_2:def 5</a></i>;<br/>
<a NAME="E12:16"/><b>then </b><i><font color="Green" title="E23">A10</font></i>: 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c3">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> 
 
<b>by </b><i><a class="ref" href="pre_topc.html#T12" title="PRE_TOPC:th.12">PRE_TOPC:12</a></i>;<br/>
<i><font color="Green" title="E24">A11</font></i>:  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c3">f</font> = 
 <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c2">P1</font></span>)</span>
<b>by </b><i><a class="txt" href="topmetr3.html#E7:16"><i><font color="Green" title="E19">A6</font></i></a>, <a class="ref" href="tops_2.html#D5" title="TOPS_2:def.5">TOPS_2:def 5</a></i>
<br/>.= 
<font color="Maroon" title="c2">P1</font>
<b>by </b><i><a class="ref" href="pre_topc.html#D10" title="PRE_TOPC:def.10">PRE_TOPC:def 10</a></i>
;<br/>
<a NAME="E14:16"/><i><font color="Green" title="E25">A12</font></i>: 
(  <a href="jordan6.html#K8" title="JORDAN6:func.8">Upper_Arc</a> <font color="Maroon" title="c1">P</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">P</font> &amp;  <a href="jordan6.html#K9" title="JORDAN6:func.9">Lower_Arc</a> <font color="Maroon" title="c1">P</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">P</font> )
 
<b>by </b><i><a class="txt" href="topmetr3.html#E1:16"><i><font color="Green" title="E14">A1</font></i></a>, <a class="ref" href="jordan6.html#T76" title="JORDAN6:th.76">JORDAN6:76</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c4">U2</font> =  <a href="jordan6.html#K8" title="JORDAN6:func.8">Upper_Arc</a> <font color="Maroon" title="c1">P</font> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c1">P</font></span>)</span> <b>by </b><i><a class="txt" href="topmetr3.html#E1:16"><i><font color="Green" title="E14">A1</font></i></a>, <a class="txt" href="topmetr3.html#E8:16"><i><font color="Green" title="E20">A7</font></i></a>, <a class="ref" href="jordan6.html#T76" title="JORDAN6:th.76">JORDAN6:76</a></i>;<br/>
<a NAME="E16:16"/>
<font color="Maroon" title="c4">U2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan6.html#K8" title="JORDAN6:func.8">Upper_Arc</a> <font color="Maroon" title="c1">P</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c1">P</font>
 
<b>by </b><i><a class="txt" href="topmetr3.html#E14:16"><i><font color="Green" title="E25">A12</font></i></a>, <a class="ref" href="xboole_1.html#T28" title="XBOOLE_1:th.28">XBOOLE_1:28</a></i>;<br/>
<a NAME="E17:16"/><b>then </b><i><font color="Green" title="E26">A13</font></i>: 
<font color="Maroon" title="c4">U2</font> is <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a>
 
<b>by </b><i><a class="txt" href="topmetr3.html#E2:16"><i><font color="Green" title="E15">A2</font></i></a>, <a class="ref" href="jordan6.html#T3" title="JORDAN6:th.3">JORDAN6:3</a>, <a class="ref" href="jordan6.html#T12" title="JORDAN6:th.12">JORDAN6:12</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c5">U3</font> =  <a href="jordan6.html#K9" title="JORDAN6:func.9">Lower_Arc</a> <font color="Maroon" title="c1">P</font> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c1">P</font></span>)</span> <b>by </b><i><a class="txt" href="topmetr3.html#E1:16"><i><font color="Green" title="E14">A1</font></i></a>, <a class="txt" href="topmetr3.html#E8:16"><i><font color="Green" title="E20">A7</font></i></a>, <a class="ref" href="jordan6.html#T76" title="JORDAN6:th.76">JORDAN6:76</a></i>;<br/>
<a NAME="E19:16"/>
<font color="Maroon" title="c5">U3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan6.html#K9" title="JORDAN6:func.9">Lower_Arc</a> <font color="Maroon" title="c1">P</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c1">P</font>
 
<b>by </b><i><a class="txt" href="topmetr3.html#E14:16"><i><font color="Green" title="E25">A12</font></i></a>, <a class="ref" href="xboole_1.html#T28" title="XBOOLE_1:th.28">XBOOLE_1:28</a></i>;<br/>
<a NAME="E20:16"/><b>then </b><i><font color="Green" title="E27">A14</font></i>: 
<font color="Maroon" title="c5">U3</font> is <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a>
 
<b>by </b><i><a class="txt" href="topmetr3.html#E3:16"><i><font color="Green" title="E16">A3</font></i></a>, <a class="ref" href="jordan6.html#T3" title="JORDAN6:th.3">JORDAN6:3</a>, <a class="ref" href="jordan6.html#T12" title="JORDAN6:th.12">JORDAN6:12</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_3"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="16_3_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:16_3_1"/>
( for <font color="Olive" title="b1">r</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>  st  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Olive" title="b1">r</font> &amp; <font color="Olive" title="b1">r</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> 1 holds <br/><font color="Maroon" title="c3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="jordan6.html#K9" title="JORDAN6:func.9">Lower_Arc</a> <font color="Maroon" title="c1">P</font> or  ex <font color="Olive" title="b1">r</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> st <br/>(  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Olive" title="b1">r</font> &amp; <font color="Olive" title="b1">r</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> 1 &amp; not <font color="Maroon" title="c3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="jordan6.html#K9" title="JORDAN6:func.9">Lower_Arc</a> <font color="Maroon" title="c1">P</font> ) )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="16_3_1_1"><b>case </b></a><a NAME="E1:16_3_1_1"/><i><font color="Green" title="E28">A15</font></i>: 
for <font color="Olive" title="b1">r</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>  st  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Olive" title="b1">r</font> &amp; <font color="Olive" title="b1">r</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> 1 holds <br/><font color="Maroon" title="c3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="jordan6.html#K9" title="JORDAN6:func.9">Lower_Arc</a> <font color="Maroon" title="c1">P</font>
 ;<br/><div class="add"><a NAME="E2:16_3_1_1"/>
<font color="Maroon" title="c2">P1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="jordan6.html#K9" title="JORDAN6:func.9">Lower_Arc</a> <font color="Maroon" title="c1">P</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_3_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c6">y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:16_3_1_1_1"/>
<font color="Maroon" title="c6">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">P1</font>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c7">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:16_3_1_1_1"/><i><font color="Green" title="E29">A16</font></i>: 
( <font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c3">f</font> &amp; <font color="Maroon" title="c6">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">x</font> )
 <b>by </b><i><a class="txt" href="topmetr3.html#E13:16"><i><font color="Green" title="E24">A11</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c8">r</font> = <font color="Maroon" title="c7">x</font> as   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> <b>by </b><i><a class="txt" href="topmetr3.html#E12:16"><i><font color="Green" title="E23">A10</font></i></a>, <a class="txt" href="topmetr3.html#E3:16_3_1_1_1"><i><font color="Green" title="E29">A16</font></i></a>, <a class="ref" href="borsuk_1.html#T83" title="BORSUK_1:th.83">BORSUK_1:83</a></i>;<br/>
<a NAME="E5:16_3_1_1_1"/><i><font color="Green" title="E30">A17</font></i>: 
(  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c8">r</font> &amp; <font color="Maroon" title="c8">r</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 )
 
<b>by </b><i><a class="txt" href="topmetr3.html#E3:16_3_1_1_1"><i><font color="Green" title="E29">A16</font></i></a>, <a class="ref" href="borsuk_1.html#T83" title="BORSUK_1:th.83">BORSUK_1:83</a>, <a class="ref" href="rcomp_1.html#T48" title="RCOMP_1:th.48">RCOMP_1:48</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_3_1_1_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="16_3_1_1_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:16_3_1_1_1_1_1"/>
( (  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c8">r</font> &amp; <font color="Maroon" title="c8">r</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> 1 ) or <font color="Maroon" title="c8">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  or <font color="Maroon" title="c8">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 )
 </span><b>by </b><i><a class="txt" href="topmetr3.html#E5:16_3_1_1_1"><i><font color="Green" title="E30">A17</font></i></a>, <a class="ref" href="real_1.html#D5" title="REAL_1:def.5">REAL_1:def 5</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="16_3_1_1_1_1_1_1"><b>case </b></a><a NAME="E1:16_3_1_1_1_1_1_1"/>
(  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c8">r</font> &amp; <font color="Maroon" title="c8">r</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> 1 )
 ;<br/><div class="add"><b>hence </b><a NAME="E2:16_3_1_1_1_1_1_1"/>
<font color="Maroon" title="c6">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="jordan6.html#K9" title="JORDAN6:func.9">Lower_Arc</a> <font color="Maroon" title="c1">P</font>
 <b>by </b><i><a class="txt" href="topmetr3.html#E1:16_3_1_1"><i><font color="Green" title="E28">A15</font></i></a>, <a class="txt" href="topmetr3.html#E3:16_3_1_1_1"><i><font color="Green" title="E29">A16</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="16_3_1_1_1_1_1_2"><b>case </b></a><a NAME="E1:16_3_1_1_1_1_1_2"/>
<font color="Maroon" title="c8">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ;<br/><div class="add"><b>hence </b><a NAME="E2:16_3_1_1_1_1_1_2"/>
<font color="Maroon" title="c6">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="jordan6.html#K9" title="JORDAN6:func.9">Lower_Arc</a> <font color="Maroon" title="c1">P</font>
 <b>by </b><i><a class="txt" href="topmetr3.html#E5:16"><i><font color="Green" title="E18">A5</font></i></a>, <a class="txt" href="topmetr3.html#E7:16"><i><font color="Green" title="E19">A6</font></i></a>, <a class="txt" href="topmetr3.html#E3:16_3_1_1_1"><i><font color="Green" title="E29">A16</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="16_3_1_1_1_1_1_3"><b>case </b></a><a NAME="E1:16_3_1_1_1_1_1_3"/>
<font color="Maroon" title="c8">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 ;<br/><div class="add"><b>hence </b><a NAME="E2:16_3_1_1_1_1_1_3"/>
<font color="Maroon" title="c6">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="jordan6.html#K9" title="JORDAN6:func.9">Lower_Arc</a> <font color="Maroon" title="c1">P</font>
 <b>by </b><i><a class="txt" href="topmetr3.html#E4:16"><i><font color="Green" title="E17">A4</font></i></a>, <a class="txt" href="topmetr3.html#E7:16"><i><font color="Green" title="E19">A6</font></i></a>, <a class="txt" href="topmetr3.html#E3:16_3_1_1_1"><i><font color="Green" title="E29">A16</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E7:16_3_1_1_1"/>
<font color="Maroon" title="c6">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="jordan6.html#K9" title="JORDAN6:func.9">Lower_Arc</a> <font color="Maroon" title="c1">P</font>
 ;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E3:16_3_1_1"/>
( <font color="Maroon" title="c2">P1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="jordan6.html#K8" title="JORDAN6:func.8">Upper_Arc</a> <font color="Maroon" title="c1">P</font> or <font color="Maroon" title="c2">P1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="jordan6.html#K9" title="JORDAN6:func.9">Lower_Arc</a> <font color="Maroon" title="c1">P</font> )
 <b>by </b><i><a class="txt" href="topmetr3.html#E1:16"><i><font color="Green" title="E14">A1</font></i></a>, <a class="txt" href="topmetr3.html#E3:16"><i><font color="Green" title="E16">A3</font></i></a>, <a class="ref" href="topmetr3.html#T18" target="_self" title="TOPMETR3:th.18">Th18</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="16_3_1_2"><b>case </b></a><a NAME="E1:16_3_1_2"/><i><font color="Green" title="E28">A18</font></i>: 
 ex <font color="Olive" title="b1">r</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> st <br/>(  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Olive" title="b1">r</font> &amp; <font color="Olive" title="b1">r</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> 1 &amp; not <font color="Maroon" title="c3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="jordan6.html#K9" title="JORDAN6:func.9">Lower_Arc</a> <font color="Maroon" title="c1">P</font> )
 ;<br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_3_1_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="16_3_1_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:16_3_1_2_1_1"/>
( for <font color="Olive" title="b1">r</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>  st  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Olive" title="b1">r</font> &amp; <font color="Olive" title="b1">r</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> 1 holds <br/><font color="Maroon" title="c3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="jordan6.html#K8" title="JORDAN6:func.8">Upper_Arc</a> <font color="Maroon" title="c1">P</font> or  ex <font color="Olive" title="b1">r</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> st <br/>(  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Olive" title="b1">r</font> &amp; <font color="Olive" title="b1">r</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> 1 &amp; not <font color="Maroon" title="c3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="jordan6.html#K8" title="JORDAN6:func.8">Upper_Arc</a> <font color="Maroon" title="c1">P</font> ) )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="16_3_1_2_1_1_1"><b>case </b></a><a NAME="E1:16_3_1_2_1_1_1"/><i><font color="Green" title="E29">A19</font></i>: 
for <font color="Olive" title="b1">r</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>  st  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Olive" title="b1">r</font> &amp; <font color="Olive" title="b1">r</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> 1 holds <br/><font color="Maroon" title="c3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="jordan6.html#K8" title="JORDAN6:func.8">Upper_Arc</a> <font color="Maroon" title="c1">P</font>
 ;<br/><div class="add"><a NAME="E2:16_3_1_2_1_1_1"/>
<font color="Maroon" title="c2">P1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="jordan6.html#K8" title="JORDAN6:func.8">Upper_Arc</a> <font color="Maroon" title="c1">P</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_3_1_2_1_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c6">y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:16_3_1_2_1_1_1_1"/>
<font color="Maroon" title="c6">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">P1</font>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c7">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:16_3_1_2_1_1_1_1"/><i><font color="Green" title="E30">A20</font></i>: 
( <font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c3">f</font> &amp; <font color="Maroon" title="c6">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">x</font> )
 <b>by </b><i><a class="txt" href="topmetr3.html#E13:16"><i><font color="Green" title="E24">A11</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c8">r</font> = <font color="Maroon" title="c7">x</font> as   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> <b>by </b><i><a class="txt" href="topmetr3.html#E12:16"><i><font color="Green" title="E23">A10</font></i></a>, <a class="txt" href="topmetr3.html#E3:16_3_1_2_1_1_1_1"><i><font color="Green" title="E30">A20</font></i></a>, <a class="ref" href="borsuk_1.html#T83" title="BORSUK_1:th.83">BORSUK_1:83</a></i>;<br/>
<a NAME="E5:16_3_1_2_1_1_1_1"/><i><font color="Green" title="E31">A21</font></i>: 
(  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c8">r</font> &amp; <font color="Maroon" title="c8">r</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 )
 
<b>by </b><i><a class="txt" href="topmetr3.html#E3:16_3_1_2_1_1_1_1"><i><font color="Green" title="E30">A20</font></i></a>, <a class="ref" href="borsuk_1.html#T83" title="BORSUK_1:th.83">BORSUK_1:83</a>, <a class="ref" href="rcomp_1.html#T48" title="RCOMP_1:th.48">RCOMP_1:48</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_3_1_2_1_1_1_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="16_3_1_2_1_1_1_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:16_3_1_2_1_1_1_1_1_1"/>
( (  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c8">r</font> &amp; <font color="Maroon" title="c8">r</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> 1 ) or <font color="Maroon" title="c8">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  or <font color="Maroon" title="c8">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 )
 </span><b>by </b><i><a class="txt" href="topmetr3.html#E5:16_3_1_2_1_1_1_1"><i><font color="Green" title="E31">A21</font></i></a>, <a class="ref" href="real_1.html#D5" title="REAL_1:def.5">REAL_1:def 5</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="16_3_1_2_1_1_1_1_1_1_1"><b>case </b></a><a NAME="E1:16_3_1_2_1_1_1_1_1_1_1"/>
(  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c8">r</font> &amp; <font color="Maroon" title="c8">r</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> 1 )
 ;<br/><div class="add"><b>hence </b><a NAME="E2:16_3_1_2_1_1_1_1_1_1_1"/>
<font color="Maroon" title="c6">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="jordan6.html#K8" title="JORDAN6:func.8">Upper_Arc</a> <font color="Maroon" title="c1">P</font>
 <b>by </b><i><a class="txt" href="topmetr3.html#E1:16_3_1_2_1_1_1"><i><font color="Green" title="E29">A19</font></i></a>, <a class="txt" href="topmetr3.html#E3:16_3_1_2_1_1_1_1"><i><font color="Green" title="E30">A20</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="16_3_1_2_1_1_1_1_1_1_2"><b>case </b></a><a NAME="E1:16_3_1_2_1_1_1_1_1_1_2"/>
<font color="Maroon" title="c8">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ;<br/><div class="add"><b>hence </b><a NAME="E2:16_3_1_2_1_1_1_1_1_1_2"/>
<font color="Maroon" title="c6">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="jordan6.html#K8" title="JORDAN6:func.8">Upper_Arc</a> <font color="Maroon" title="c1">P</font>
 <b>by </b><i><a class="txt" href="topmetr3.html#E5:16"><i><font color="Green" title="E18">A5</font></i></a>, <a class="txt" href="topmetr3.html#E7:16"><i><font color="Green" title="E19">A6</font></i></a>, <a class="txt" href="topmetr3.html#E3:16_3_1_2_1_1_1_1"><i><font color="Green" title="E30">A20</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="16_3_1_2_1_1_1_1_1_1_3"><b>case </b></a><a NAME="E1:16_3_1_2_1_1_1_1_1_1_3"/>
<font color="Maroon" title="c8">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 ;<br/><div class="add"><b>hence </b><a NAME="E2:16_3_1_2_1_1_1_1_1_1_3"/>
<font color="Maroon" title="c6">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="jordan6.html#K8" title="JORDAN6:func.8">Upper_Arc</a> <font color="Maroon" title="c1">P</font>
 <b>by </b><i><a class="txt" href="topmetr3.html#E4:16"><i><font color="Green" title="E17">A4</font></i></a>, <a class="txt" href="topmetr3.html#E7:16"><i><font color="Green" title="E19">A6</font></i></a>, <a class="txt" href="topmetr3.html#E3:16_3_1_2_1_1_1_1"><i><font color="Green" title="E30">A20</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E7:16_3_1_2_1_1_1_1"/>
<font color="Maroon" title="c6">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="jordan6.html#K8" title="JORDAN6:func.8">Upper_Arc</a> <font color="Maroon" title="c1">P</font>
 ;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E3:16_3_1_2_1_1_1"/>
( <font color="Maroon" title="c2">P1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="jordan6.html#K8" title="JORDAN6:func.8">Upper_Arc</a> <font color="Maroon" title="c1">P</font> or <font color="Maroon" title="c2">P1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="jordan6.html#K9" title="JORDAN6:func.9">Lower_Arc</a> <font color="Maroon" title="c1">P</font> )
 <b>by </b><i><a class="txt" href="topmetr3.html#E1:16"><i><font color="Green" title="E14">A1</font></i></a>, <a class="txt" href="topmetr3.html#E2:16"><i><font color="Green" title="E15">A2</font></i></a>, <a class="ref" href="jordan6.html#T59" title="JORDAN6:th.59">JORDAN6:59</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="16_3_1_2_1_1_2"><b>case </b></a><a NAME="E1:16_3_1_2_1_1_2"/>
 ex <font color="Olive" title="b1">r</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> st <br/>(  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Olive" title="b1">r</font> &amp; <font color="Olive" title="b1">r</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> 1 &amp; not <font color="Maroon" title="c3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="jordan6.html#K8" title="JORDAN6:func.8">Upper_Arc</a> <font color="Maroon" title="c1">P</font> )
 ;<br/><div class="add"><b>then consider </b><font color="Maroon" title="c6">r2</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a><b> such that </b><br/><a NAME="E3:16_3_1_2_1_1_2"/><i><font color="Green" title="E29">A22</font></i>: 
(  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c6">r2</font> &amp; <font color="Maroon" title="c6">r2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> 1 &amp; not <font color="Maroon" title="c3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">r2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="jordan6.html#K8" title="JORDAN6:func.8">Upper_Arc</a> <font color="Maroon" title="c1">P</font> )
 ;<br/><b>consider </b><font color="Maroon" title="c7">r1</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a><b> such that </b><br/><a NAME="E5:16_3_1_2_1_1_2"/><i><font color="Green" title="E30">A23</font></i>: 
(  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c7">r1</font> &amp; <font color="Maroon" title="c7">r1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> 1 &amp; not <font color="Maroon" title="c3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">r1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="jordan6.html#K9" title="JORDAN6:func.9">Lower_Arc</a> <font color="Maroon" title="c1">P</font> )
 <b>by </b><i><a class="txt" href="topmetr3.html#E1:16_3_1_2"><i><font color="Green" title="E28">A18</font></i></a></i>;<br/><a NAME="E6:16_3_1_2_1_1_2"/>
<font color="Maroon" title="c6">r2</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>
 <b>by </b><i><a class="txt" href="topmetr3.html#E3:16_3_1_2_1_1_2"><i><font color="Green" title="E29">A22</font></i></a>, <a class="ref" href="rcomp_1.html#T48" title="RCOMP_1:th.48">RCOMP_1:48</a></i>;<br/><a NAME="E7:16_3_1_2_1_1_2"/><b>then </b>
<font color="Maroon" title="c3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">r2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c3">f</font>
 <b>by </b><i><a class="txt" href="topmetr3.html#E12:16"><i><font color="Green" title="E23">A10</font></i></a>, <a class="ref" href="borsuk_1.html#T83" title="BORSUK_1:th.83">BORSUK_1:83</a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/><a NAME="E8:16_3_1_2_1_1_2"/><b>then </b><i><font color="Green" title="E31">A24</font></i>: 
<font color="Maroon" title="c3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">r2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="jordan6.html#K9" title="JORDAN6:func.9">Lower_Arc</a> <font color="Maroon" title="c1">P</font>
 <b>by </b><i><a class="txt" href="topmetr3.html#E1:16"><i><font color="Green" title="E14">A1</font></i></a>, <a class="txt" href="topmetr3.html#E3:16"><i><font color="Green" title="E16">A3</font></i></a>, <a class="txt" href="topmetr3.html#E13:16"><i><font color="Green" title="E24">A11</font></i></a>, <a class="txt" href="topmetr3.html#E3:16_3_1_2_1_1_2"><i><font color="Green" title="E29">A22</font></i></a>, <a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">XBOOLE_0:def 2</a></i>;<br/><a NAME="E9:16_3_1_2_1_1_2"/>
<font color="Maroon" title="c7">r1</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>
 <b>by </b><i><a class="txt" href="topmetr3.html#E5:16_3_1_2_1_1_2"><i><font color="Green" title="E30">A23</font></i></a>, <a class="ref" href="rcomp_1.html#T48" title="RCOMP_1:th.48">RCOMP_1:48</a></i>;<br/><a NAME="E10:16_3_1_2_1_1_2"/><b>then </b><i><font color="Green" title="E32">A25</font></i>: 
<font color="Maroon" title="c3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">r1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c3">f</font>
 <b>by </b><i><a class="txt" href="topmetr3.html#E12:16"><i><font color="Green" title="E23">A10</font></i></a>, <a class="ref" href="borsuk_1.html#T83" title="BORSUK_1:th.83">BORSUK_1:83</a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/><a NAME="E11:16_3_1_2_1_1_2"/><b>then </b><i><font color="Green" title="E33">A26</font></i>: 
<font color="Maroon" title="c3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">r1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="jordan6.html#K8" title="JORDAN6:func.8">Upper_Arc</a> <font color="Maroon" title="c1">P</font>
 <b>by </b><i><a class="txt" href="topmetr3.html#E1:16"><i><font color="Green" title="E14">A1</font></i></a>, <a class="txt" href="topmetr3.html#E3:16"><i><font color="Green" title="E16">A3</font></i></a>, <a class="txt" href="topmetr3.html#E13:16"><i><font color="Green" title="E24">A11</font></i></a>, <a class="txt" href="topmetr3.html#E5:16_3_1_2_1_1_2"><i><font color="Green" title="E30">A23</font></i></a>, <a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">XBOOLE_0:def 2</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_3_1_2_1_1_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="16_3_1_2_1_1_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:16_3_1_2_1_1_2_1_1"/>
( <font color="Maroon" title="c7">r1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">r2</font> or <font color="Maroon" title="c7">r1</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c6">r2</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="16_3_1_2_1_1_2_1_1_1"><b>case </b></a><a NAME="E1:16_3_1_2_1_1_2_1_1_1"/><i><font color="Green" title="E34">A27</font></i>: 
<font color="Maroon" title="c7">r1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">r2</font>
 ;<br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_3_1_2_1_1_2_1_1_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="16_3_1_2_1_1_2_1_1_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:16_3_1_2_1_1_2_1_1_1_1_1"/>
( <font color="Maroon" title="c7">r1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">r2</font> or <font color="Maroon" title="c7">r1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c6">r2</font> )
 </span><b>by </b><i><a class="txt" href="topmetr3.html#E1:16_3_1_2_1_1_2_1_1_1"><i><font color="Green" title="E34">A27</font></i></a>, <a class="ref" href="real_1.html#D5" title="REAL_1:def.5">REAL_1:def 5</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="16_3_1_2_1_1_2_1_1_1_1_1_1"><b>case </b></a><a NAME="E1:16_3_1_2_1_1_2_1_1_1_1_1_1"/>
<font color="Maroon" title="c7">r1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">r2</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:16_3_1_2_1_1_2_1_1_1_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="topmetr3.html#E1:16"><i><font color="Green" title="E14">A1</font></i></a>, <a class="txt" href="topmetr3.html#E3:16"><i><font color="Green" title="E16">A3</font></i></a>, <a class="txt" href="topmetr3.html#E13:16"><i><font color="Green" title="E24">A11</font></i></a>, <a class="txt" href="topmetr3.html#E3:16_3_1_2_1_1_2"><i><font color="Green" title="E29">A22</font></i></a>, <a class="txt" href="topmetr3.html#E5:16_3_1_2_1_1_2"><i><font color="Green" title="E30">A23</font></i></a>, <a class="txt" href="topmetr3.html#E10:16_3_1_2_1_1_2"><i><font color="Green" title="E32">A25</font></i></a>, <a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">XBOOLE_0:def 2</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="16_3_1_2_1_1_2_1_1_1_1_1_2"><b>case </b></a><a NAME="E1:16_3_1_2_1_1_2_1_1_1_1_1_2"/><i><font color="Green" title="E35">A28</font></i>: 
<font color="Maroon" title="c7">r1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c6">r2</font>
 ;<br/><div class="add"><i><font color="Green" title="E36">A29</font></i>: the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c1">P</font></span>)</span> = 
 <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c1">P</font></span>)</span>
<b>by </b><i><a class="ref" href="pre_topc.html#T12" title="PRE_TOPC:th.12">PRE_TOPC:12</a></i>
<br/>.= 
<font color="Maroon" title="c1">P</font>
<b>by </b><i><a class="ref" href="pre_topc.html#D10" title="PRE_TOPC:def.10">PRE_TOPC:def 10</a></i>
;<br/>the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c2">P1</font></span>)</span> = 
 <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c2">P1</font></span>)</span>
<b>by </b><i><a class="ref" href="pre_topc.html#T12" title="PRE_TOPC:th.12">PRE_TOPC:12</a></i>
<br/>.= 
<font color="Maroon" title="c2">P1</font>
<b>by </b><i><a class="ref" href="pre_topc.html#D10" title="PRE_TOPC:def.10">PRE_TOPC:def 10</a></i>
;<br/><a NAME="E4:16_3_1_2_1_1_2_1_1_1_1_1_2"/><b>then </b>
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c3">f</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c1">P</font></span>)</span>
 <b>by </b><i><a class="txt" href="topmetr3.html#E1:16"><i><font color="Green" title="E14">A1</font></i></a>, <a class="txt" href="topmetr3.html#E2:16_3_1_2_1_1_2_1_1_1_1_1_2"><i><font color="Green" title="E36">A29</font></i></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c8">g</font> = <font color="Maroon" title="c3">f</font> as   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> ,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c1">P</font></span>)</span> <b>by </b><i><a class="txt" href="topmetr3.html#E12:16"><i><font color="Green" title="E23">A10</font></i></a>, <a class="ref" href="funct_2.html#T4" title="FUNCT_2:th.4">FUNCT_2:4</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c9">Q</font> = <font color="Maroon" title="c1">P</font> as  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K14" title="EUCLID:func.14">Euclid</a> 2</span>)</span> <b>by </b><i><a class="ref" href="topreal3.html#T13" title="TOPREAL3:th.13">TOPREAL3:13</a></i>;<br/><a NAME="E7:16_3_1_2_1_1_2_1_1_1_1_1_2"/>
<font color="Maroon" title="c1">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">P1</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c1">P</font>
 <b>by </b><i><a class="txt" href="topmetr3.html#E1:16"><i><font color="Green" title="E14">A1</font></i></a>, <a class="ref" href="xboole_1.html#T12" title="XBOOLE_1:th.12">XBOOLE_1:12</a></i>;<br/><a NAME="E8:16_3_1_2_1_1_2_1_1_1_1_1_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c2">P1</font> is    <a href="pre_topc.html#M1" title="PRE_TOPC:mode.1">SubSpace</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c1">P</font>
 <b>by </b><i><a class="ref" href="topmetr.html#T5" title="TOPMETR:th.5">TOPMETR:5</a></i>;<br/><a NAME="E9:16_3_1_2_1_1_2_1_1_1_1_1_2"/><b>then </b><i><font color="Green" title="E37">A30</font></i>: 
<font color="Maroon" title="c8">g</font> is <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a>
 <b>by </b><i><a class="txt" href="topmetr3.html#E10:16"><i><font color="Green" title="E22">A9</font></i></a>, <a class="ref" href="topmetr.html#T7" title="TOPMETR:th.7">TOPMETR:7</a></i>;<br/><a NAME="E10:16_3_1_2_1_1_2_1_1_1_1_1_2"/><i><font color="Green" title="E38">A31</font></i>: 
<font color="Maroon" title="c4">U2</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c5">U3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K14" title="EUCLID:func.14">Euclid</a> 2</span>)</span> <a href="topmetr.html#K1" title="TOPMETR:func.1">|</a> <font color="Maroon" title="c9">Q</font></span>)</span>
 <b>by </b><i><a class="txt" href="topmetr3.html#E3:16"><i><font color="Green" title="E16">A3</font></i></a>, <a class="ref" href="topmetr.html#D2" title="TOPMETR:def.2">TOPMETR:def 2</a></i>;<br/><a NAME="E11:16_3_1_2_1_1_2_1_1_1_1_1_2"/>
<span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c1">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pcomps_1.html#K4" title="PCOMPS_1:func.4">TopSpaceMetr</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K14" title="EUCLID:func.14">Euclid</a> 2</span>)</span> <a href="topmetr.html#K1" title="TOPMETR:func.1">|</a> <font color="Maroon" title="c9">Q</font></span>)</span>
 <b>by </b><i><a class="ref" href="topmetr.html#T20" title="TOPMETR:th.20">TOPMETR:20</a></i>;<br/><b>then consider </b><font color="Maroon" title="c10">r0</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a><b> such that </b><br/><a NAME="E13:16_3_1_2_1_1_2_1_1_1_1_1_2"/><i><font color="Green" title="E39">A32</font></i>: 
( <font color="Maroon" title="c7">r1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c10">r0</font> &amp; <font color="Maroon" title="c10">r0</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">r2</font> &amp; <font color="Maroon" title="c8">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">r0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">U2</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c5">U3</font> )
 <b>by </b><i><a class="txt" href="topmetr3.html#E17:16"><i><font color="Green" title="E26">A13</font></i></a>, <a class="txt" href="topmetr3.html#E20:16"><i><font color="Green" title="E27">A14</font></i></a>, <a class="txt" href="topmetr3.html#E3:16_3_1_2_1_1_2"><i><font color="Green" title="E29">A22</font></i></a>, <a class="txt" href="topmetr3.html#E5:16_3_1_2_1_1_2"><i><font color="Green" title="E30">A23</font></i></a>, <a class="txt" href="topmetr3.html#E8:16_3_1_2_1_1_2"><i><font color="Green" title="E31">A24</font></i></a>, <a class="txt" href="topmetr3.html#E11:16_3_1_2_1_1_2"><i><font color="Green" title="E33">A26</font></i></a>, <a class="txt" href="topmetr3.html#E1:16_3_1_2_1_1_2_1_1_1_1_1_2"><i><font color="Green" title="E35">A28</font></i></a>, <a class="txt" href="topmetr3.html#E9:16_3_1_2_1_1_2_1_1_1_1_1_2"><i><font color="Green" title="E37">A30</font></i></a>, <a class="txt" href="topmetr3.html#E10:16_3_1_2_1_1_2_1_1_1_1_1_2"><i><font color="Green" title="E38">A31</font></i></a>, <a class="ref" href="topmetr3.html#T17" target="_self" title="TOPMETR3:th.17">Th17</a></i>;<br/><a NAME="E14:16_3_1_2_1_1_2_1_1_1_1_1_2"/>
<font color="Maroon" title="c10">r0</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> 1
 <b>by </b><i><a class="txt" href="topmetr3.html#E3:16_3_1_2_1_1_2"><i><font color="Green" title="E29">A22</font></i></a>, <a class="txt" href="topmetr3.html#E13:16_3_1_2_1_1_2_1_1_1_1_1_2"><i><font color="Green" title="E39">A32</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E15:16_3_1_2_1_1_2_1_1_1_1_1_2"/><b>then </b><i><font color="Green" title="E40">A33</font></i>: 
<font color="Maroon" title="c10">r0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c3">f</font>
 <b>by </b><i><a class="txt" href="topmetr3.html#E12:16"><i><font color="Green" title="E23">A10</font></i></a>, <a class="txt" href="topmetr3.html#E5:16_3_1_2_1_1_2"><i><font color="Green" title="E30">A23</font></i></a>, <a class="txt" href="topmetr3.html#E13:16_3_1_2_1_1_2_1_1_1_1_1_2"><i><font color="Green" title="E39">A32</font></i></a>, <a class="ref" href="borsuk_1.html#T83" title="BORSUK_1:th.83">BORSUK_1:83</a>, <a class="ref" href="rcomp_1.html#T48" title="RCOMP_1:th.48">RCOMP_1:48</a></i>;<br/><a NAME="E16:16_3_1_2_1_1_2_1_1_1_1_1_2"/><i><font color="Green" title="E41">A34</font></i>: 
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c3">f</font>
 <b>by </b><i><a class="txt" href="topmetr3.html#E12:16"><i><font color="Green" title="E23">A10</font></i></a>, <a class="ref" href="borsuk_1.html#T83" title="BORSUK_1:th.83">BORSUK_1:83</a>, <a class="ref" href="rcomp_1.html#T48" title="RCOMP_1:th.48">RCOMP_1:48</a></i>;<br/><a NAME="E17:16_3_1_2_1_1_2_1_1_1_1_1_2"/><i><font color="Green" title="E42">A35</font></i>: 
1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c3">f</font>
 <b>by </b><i><a class="txt" href="topmetr3.html#E12:16"><i><font color="Green" title="E23">A10</font></i></a>, <a class="ref" href="borsuk_1.html#T83" title="BORSUK_1:th.83">BORSUK_1:83</a>, <a class="ref" href="rcomp_1.html#T48" title="RCOMP_1:th.48">RCOMP_1:48</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_3_1_2_1_1_2_1_1_1_1_1_2_3"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="16_3_1_2_1_1_2_1_1_1_1_1_2_3_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:16_3_1_2_1_1_2_1_1_1_1_1_2_3_1"/>
( <font color="Maroon" title="c8">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">r0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pscomp_1.html#K25" title="PSCOMP_1:func.25">W-min</a> <font color="Maroon" title="c1">P</font> or <font color="Maroon" title="c8">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">r0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pscomp_1.html#K29" title="PSCOMP_1:func.29">E-max</a> <font color="Maroon" title="c1">P</font> )
 </span><b>by </b><i><a class="txt" href="topmetr3.html#E3:16"><i><font color="Green" title="E16">A3</font></i></a>, <a class="txt" href="topmetr3.html#E13:16_3_1_2_1_1_2_1_1_1_1_1_2"><i><font color="Green" title="E39">A32</font></i></a>, <a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="16_3_1_2_1_1_2_1_1_1_1_1_2_3_1_1"><b>case </b></a><a NAME="E1:16_3_1_2_1_1_2_1_1_1_1_1_2_3_1_1"/>
<font color="Maroon" title="c8">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">r0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pscomp_1.html#K25" title="PSCOMP_1:func.25">W-min</a> <font color="Maroon" title="c1">P</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:16_3_1_2_1_1_2_1_1_1_1_1_2_3_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="topmetr3.html#E7:16"><i><font color="Green" title="E19">A6</font></i></a>, <a class="txt" href="topmetr3.html#E9:16"><i><font color="Green" title="E21">A8</font></i></a>, <a class="txt" href="topmetr3.html#E5:16_3_1_2_1_1_2"><i><font color="Green" title="E30">A23</font></i></a>, <a class="txt" href="topmetr3.html#E13:16_3_1_2_1_1_2_1_1_1_1_1_2"><i><font color="Green" title="E39">A32</font></i></a>, <a class="txt" href="topmetr3.html#E15:16_3_1_2_1_1_2_1_1_1_1_1_2"><i><font color="Green" title="E40">A33</font></i></a>, <a class="txt" href="topmetr3.html#E16:16_3_1_2_1_1_2_1_1_1_1_1_2"><i><font color="Green" title="E41">A34</font></i></a>, <a class="ref" href="funct_1.html#D8" title="FUNCT_1:def.8">FUNCT_1:def 8</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="16_3_1_2_1_1_2_1_1_1_1_1_2_3_1_2"><b>case </b></a><a NAME="E1:16_3_1_2_1_1_2_1_1_1_1_1_2_3_1_2"/>
<font color="Maroon" title="c8">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">r0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pscomp_1.html#K29" title="PSCOMP_1:func.29">E-max</a> <font color="Maroon" title="c1">P</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:16_3_1_2_1_1_2_1_1_1_1_1_2_3_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="topmetr3.html#E7:16"><i><font color="Green" title="E19">A6</font></i></a>, <a class="txt" href="topmetr3.html#E9:16"><i><font color="Green" title="E21">A8</font></i></a>, <a class="txt" href="topmetr3.html#E3:16_3_1_2_1_1_2"><i><font color="Green" title="E29">A22</font></i></a>, <a class="txt" href="topmetr3.html#E13:16_3_1_2_1_1_2_1_1_1_1_1_2"><i><font color="Green" title="E39">A32</font></i></a>, <a class="txt" href="topmetr3.html#E15:16_3_1_2_1_1_2_1_1_1_1_1_2"><i><font color="Green" title="E40">A33</font></i></a>, <a class="txt" href="topmetr3.html#E17:16_3_1_2_1_1_2_1_1_1_1_1_2"><i><font color="Green" title="E42">A35</font></i></a>, <a class="ref" href="funct_1.html#D8" title="FUNCT_1:def.8">FUNCT_1:def 8</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E19:16_3_1_2_1_1_2_1_1_1_1_1_2"/>
contradiction
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E3:16_3_1_2_1_1_2_1_1_1"/>
contradiction
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="16_3_1_2_1_1_2_1_1_2"><b>case </b></a><a NAME="E1:16_3_1_2_1_1_2_1_1_2"/><i><font color="Green" title="E34">A36</font></i>: 
<font color="Maroon" title="c7">r1</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c6">r2</font>
 ;<br/><div class="add"><i><font color="Green" title="E35">A37</font></i>: the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c1">P</font></span>)</span> = 
 <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c1">P</font></span>)</span>
<b>by </b><i><a class="ref" href="pre_topc.html#T12" title="PRE_TOPC:th.12">PRE_TOPC:12</a></i>
<br/>.= 
<font color="Maroon" title="c1">P</font>
<b>by </b><i><a class="ref" href="pre_topc.html#D10" title="PRE_TOPC:def.10">PRE_TOPC:def 10</a></i>
;<br/>the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c2">P1</font></span>)</span> = 
 <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c2">P1</font></span>)</span>
<b>by </b><i><a class="ref" href="pre_topc.html#T12" title="PRE_TOPC:th.12">PRE_TOPC:12</a></i>
<br/>.= 
<font color="Maroon" title="c2">P1</font>
<b>by </b><i><a class="ref" href="pre_topc.html#D10" title="PRE_TOPC:def.10">PRE_TOPC:def 10</a></i>
;<br/><a NAME="E4:16_3_1_2_1_1_2_1_1_2"/><b>then </b>
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c3">f</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c1">P</font></span>)</span>
 <b>by </b><i><a class="txt" href="topmetr3.html#E1:16"><i><font color="Green" title="E14">A1</font></i></a>, <a class="txt" href="topmetr3.html#E2:16_3_1_2_1_1_2_1_1_2"><i><font color="Green" title="E35">A37</font></i></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c8">g</font> = <font color="Maroon" title="c3">f</font> as   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> ,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c1">P</font></span>)</span> <b>by </b><i><a class="txt" href="topmetr3.html#E12:16"><i><font color="Green" title="E23">A10</font></i></a>, <a class="ref" href="funct_2.html#T4" title="FUNCT_2:th.4">FUNCT_2:4</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c9">Q</font> = <font color="Maroon" title="c1">P</font> as  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K14" title="EUCLID:func.14">Euclid</a> 2</span>)</span> <b>by </b><i><a class="ref" href="topreal3.html#T13" title="TOPREAL3:th.13">TOPREAL3:13</a></i>;<br/><a NAME="E7:16_3_1_2_1_1_2_1_1_2"/>
<font color="Maroon" title="c1">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">P1</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c1">P</font>
 <b>by </b><i><a class="txt" href="topmetr3.html#E1:16"><i><font color="Green" title="E14">A1</font></i></a>, <a class="ref" href="xboole_1.html#T12" title="XBOOLE_1:th.12">XBOOLE_1:12</a></i>;<br/><a NAME="E8:16_3_1_2_1_1_2_1_1_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c2">P1</font> is    <a href="pre_topc.html#M1" title="PRE_TOPC:mode.1">SubSpace</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c1">P</font>
 <b>by </b><i><a class="ref" href="topmetr.html#T5" title="TOPMETR:th.5">TOPMETR:5</a></i>;<br/><a NAME="E9:16_3_1_2_1_1_2_1_1_2"/><b>then </b><i><font color="Green" title="E36">A38</font></i>: 
<font color="Maroon" title="c8">g</font> is <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a>
 <b>by </b><i><a class="txt" href="topmetr3.html#E10:16"><i><font color="Green" title="E22">A9</font></i></a>, <a class="ref" href="topmetr.html#T7" title="TOPMETR:th.7">TOPMETR:7</a></i>;<br/><a NAME="E10:16_3_1_2_1_1_2_1_1_2"/><i><font color="Green" title="E37">A39</font></i>: 
<font color="Maroon" title="c4">U2</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c5">U3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K14" title="EUCLID:func.14">Euclid</a> 2</span>)</span> <a href="topmetr.html#K1" title="TOPMETR:func.1">|</a> <font color="Maroon" title="c9">Q</font></span>)</span>
 <b>by </b><i><a class="txt" href="topmetr3.html#E3:16"><i><font color="Green" title="E16">A3</font></i></a>, <a class="ref" href="topmetr.html#D2" title="TOPMETR:def.2">TOPMETR:def 2</a></i>;<br/><a NAME="E11:16_3_1_2_1_1_2_1_1_2"/>
<span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c1">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pcomps_1.html#K4" title="PCOMPS_1:func.4">TopSpaceMetr</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K14" title="EUCLID:func.14">Euclid</a> 2</span>)</span> <a href="topmetr.html#K1" title="TOPMETR:func.1">|</a> <font color="Maroon" title="c9">Q</font></span>)</span>
 <b>by </b><i><a class="ref" href="topmetr.html#T20" title="TOPMETR:th.20">TOPMETR:20</a></i>;<br/><b>then consider </b><font color="Maroon" title="c10">r0</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a><b> such that </b><br/><a NAME="E13:16_3_1_2_1_1_2_1_1_2"/><i><font color="Green" title="E38">A40</font></i>: 
( <font color="Maroon" title="c6">r2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c10">r0</font> &amp; <font color="Maroon" title="c10">r0</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c7">r1</font> &amp; <font color="Maroon" title="c8">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">r0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">U2</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c5">U3</font> )
 <b>by </b><i><a class="txt" href="topmetr3.html#E17:16"><i><font color="Green" title="E26">A13</font></i></a>, <a class="txt" href="topmetr3.html#E20:16"><i><font color="Green" title="E27">A14</font></i></a>, <a class="txt" href="topmetr3.html#E3:16_3_1_2_1_1_2"><i><font color="Green" title="E29">A22</font></i></a>, <a class="txt" href="topmetr3.html#E5:16_3_1_2_1_1_2"><i><font color="Green" title="E30">A23</font></i></a>, <a class="txt" href="topmetr3.html#E8:16_3_1_2_1_1_2"><i><font color="Green" title="E31">A24</font></i></a>, <a class="txt" href="topmetr3.html#E11:16_3_1_2_1_1_2"><i><font color="Green" title="E33">A26</font></i></a>, <a class="txt" href="topmetr3.html#E1:16_3_1_2_1_1_2_1_1_2"><i><font color="Green" title="E34">A36</font></i></a>, <a class="txt" href="topmetr3.html#E9:16_3_1_2_1_1_2_1_1_2"><i><font color="Green" title="E36">A38</font></i></a>, <a class="txt" href="topmetr3.html#E10:16_3_1_2_1_1_2_1_1_2"><i><font color="Green" title="E37">A39</font></i></a>, <a class="ref" href="topmetr3.html#T17" target="_self" title="TOPMETR3:th.17">Th17</a></i>;<br/><a NAME="E14:16_3_1_2_1_1_2_1_1_2"/>
<font color="Maroon" title="c10">r0</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> 1
 <b>by </b><i><a class="txt" href="topmetr3.html#E5:16_3_1_2_1_1_2"><i><font color="Green" title="E30">A23</font></i></a>, <a class="txt" href="topmetr3.html#E13:16_3_1_2_1_1_2_1_1_2"><i><font color="Green" title="E38">A40</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E15:16_3_1_2_1_1_2_1_1_2"/><b>then </b><i><font color="Green" title="E39">A41</font></i>: 
<font color="Maroon" title="c10">r0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c3">f</font>
 <b>by </b><i><a class="txt" href="topmetr3.html#E12:16"><i><font color="Green" title="E23">A10</font></i></a>, <a class="txt" href="topmetr3.html#E3:16_3_1_2_1_1_2"><i><font color="Green" title="E29">A22</font></i></a>, <a class="txt" href="topmetr3.html#E13:16_3_1_2_1_1_2_1_1_2"><i><font color="Green" title="E38">A40</font></i></a>, <a class="ref" href="borsuk_1.html#T83" title="BORSUK_1:th.83">BORSUK_1:83</a>, <a class="ref" href="rcomp_1.html#T48" title="RCOMP_1:th.48">RCOMP_1:48</a></i>;<br/><a NAME="E16:16_3_1_2_1_1_2_1_1_2"/><i><font color="Green" title="E40">A42</font></i>: 
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c3">f</font>
 <b>by </b><i><a class="txt" href="topmetr3.html#E12:16"><i><font color="Green" title="E23">A10</font></i></a>, <a class="ref" href="borsuk_1.html#T83" title="BORSUK_1:th.83">BORSUK_1:83</a>, <a class="ref" href="rcomp_1.html#T48" title="RCOMP_1:th.48">RCOMP_1:48</a></i>;<br/><a NAME="E17:16_3_1_2_1_1_2_1_1_2"/><i><font color="Green" title="E41">A43</font></i>: 
1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c3">f</font>
 <b>by </b><i><a class="txt" href="topmetr3.html#E12:16"><i><font color="Green" title="E23">A10</font></i></a>, <a class="ref" href="borsuk_1.html#T83" title="BORSUK_1:th.83">BORSUK_1:83</a>, <a class="ref" href="rcomp_1.html#T48" title="RCOMP_1:th.48">RCOMP_1:48</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_3_1_2_1_1_2_1_1_2_3"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="16_3_1_2_1_1_2_1_1_2_3_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:16_3_1_2_1_1_2_1_1_2_3_1"/>
( <font color="Maroon" title="c8">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">r0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pscomp_1.html#K25" title="PSCOMP_1:func.25">W-min</a> <font color="Maroon" title="c1">P</font> or <font color="Maroon" title="c8">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">r0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pscomp_1.html#K29" title="PSCOMP_1:func.29">E-max</a> <font color="Maroon" title="c1">P</font> )
 </span><b>by </b><i><a class="txt" href="topmetr3.html#E3:16"><i><font color="Green" title="E16">A3</font></i></a>, <a class="txt" href="topmetr3.html#E13:16_3_1_2_1_1_2_1_1_2"><i><font color="Green" title="E38">A40</font></i></a>, <a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="16_3_1_2_1_1_2_1_1_2_3_1_1"><b>case </b></a><a NAME="E1:16_3_1_2_1_1_2_1_1_2_3_1_1"/>
<font color="Maroon" title="c8">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">r0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pscomp_1.html#K25" title="PSCOMP_1:func.25">W-min</a> <font color="Maroon" title="c1">P</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:16_3_1_2_1_1_2_1_1_2_3_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="topmetr3.html#E7:16"><i><font color="Green" title="E19">A6</font></i></a>, <a class="txt" href="topmetr3.html#E9:16"><i><font color="Green" title="E21">A8</font></i></a>, <a class="txt" href="topmetr3.html#E3:16_3_1_2_1_1_2"><i><font color="Green" title="E29">A22</font></i></a>, <a class="txt" href="topmetr3.html#E13:16_3_1_2_1_1_2_1_1_2"><i><font color="Green" title="E38">A40</font></i></a>, <a class="txt" href="topmetr3.html#E15:16_3_1_2_1_1_2_1_1_2"><i><font color="Green" title="E39">A41</font></i></a>, <a class="txt" href="topmetr3.html#E16:16_3_1_2_1_1_2_1_1_2"><i><font color="Green" title="E40">A42</font></i></a>, <a class="ref" href="funct_1.html#D8" title="FUNCT_1:def.8">FUNCT_1:def 8</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="16_3_1_2_1_1_2_1_1_2_3_1_2"><b>case </b></a><a NAME="E1:16_3_1_2_1_1_2_1_1_2_3_1_2"/>
<font color="Maroon" title="c8">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">r0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pscomp_1.html#K29" title="PSCOMP_1:func.29">E-max</a> <font color="Maroon" title="c1">P</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:16_3_1_2_1_1_2_1_1_2_3_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="topmetr3.html#E7:16"><i><font color="Green" title="E19">A6</font></i></a>, <a class="txt" href="topmetr3.html#E9:16"><i><font color="Green" title="E21">A8</font></i></a>, <a class="txt" href="topmetr3.html#E5:16_3_1_2_1_1_2"><i><font color="Green" title="E30">A23</font></i></a>, <a class="txt" href="topmetr3.html#E13:16_3_1_2_1_1_2_1_1_2"><i><font color="Green" title="E38">A40</font></i></a>, <a class="txt" href="topmetr3.html#E15:16_3_1_2_1_1_2_1_1_2"><i><font color="Green" title="E39">A41</font></i></a>, <a class="txt" href="topmetr3.html#E17:16_3_1_2_1_1_2_1_1_2"><i><font color="Green" title="E41">A43</font></i></a>, <a class="ref" href="funct_1.html#D8" title="FUNCT_1:def.8">FUNCT_1:def 8</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E19:16_3_1_2_1_1_2_1_1_2"/>
contradiction
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E13:16_3_1_2_1_1_2"/>
contradiction
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E3:16_3_1_2"/>
( <font color="Maroon" title="c2">P1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="jordan6.html#K8" title="JORDAN6:func.8">Upper_Arc</a> <font color="Maroon" title="c1">P</font> or <font color="Maroon" title="c2">P1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="jordan6.html#K9" title="JORDAN6:func.9">Lower_Arc</a> <font color="Maroon" title="c1">P</font> )
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E22:16"/>
( <font color="Maroon" title="c2">P1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="jordan6.html#K8" title="JORDAN6:func.8">Upper_Arc</a> <font color="Maroon" title="c1">P</font> or <font color="Maroon" title="c2">P1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="jordan6.html#K9" title="JORDAN6:func.9">Lower_Arc</a> <font color="Maroon" title="c1">P</font> )
 ;<br/>


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