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<b>let </b><font color="Maroon" title="c1">P1</font>, <font color="Maroon" title="c2">P2</font>, <font color="Maroon" title="c3">P3</font> be   <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>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">p1</font>, <font color="Olive" title="b2">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>  st <font color="Maroon" title="c1">P1</font> <a href="topreal1.html#R1" title="TOPREAL1:pred.1">is_an_arc_of</a> <font color="Olive" title="b1">p1</font>,<font color="Olive" title="b2">p2</font> &amp; <font color="Maroon" title="c2">P2</font> <a href="topreal1.html#R1" title="TOPREAL1:pred.1">is_an_arc_of</a> <font color="Olive" title="b1">p1</font>,<font color="Olive" title="b2">p2</font> &amp; <font color="Maroon" title="c3">P3</font> <a href="topreal1.html#R1" title="TOPREAL1:pred.1">is_an_arc_of</a> <font color="Olive" title="b1">p1</font>,<font color="Olive" title="b2">p2</font> &amp; <font color="Maroon" title="c2">P2</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c3">P3</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"><font color="Olive" title="b1">p1</font>,<font color="Olive" title="b2">p2</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> &amp; <font color="Maroon" title="c1">P1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">P2</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c3">P3</font> &amp; not <font color="Maroon" title="c1">P1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">P2</font> holds <br/><font color="Maroon" title="c1">P1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">P3</font></span><br/><b>let </b><font color="Maroon" title="c4">p1</font>, <font color="Maroon" title="c5">p2</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:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c1">P1</font> <a href="topreal1.html#R1" title="TOPREAL1:pred.1">is_an_arc_of</a> <font color="Maroon" title="c4">p1</font>,<font color="Maroon" title="c5">p2</font> &amp; <font color="Maroon" title="c2">P2</font> <a href="topreal1.html#R1" title="TOPREAL1:pred.1">is_an_arc_of</a> <font color="Maroon" title="c4">p1</font>,<font color="Maroon" title="c5">p2</font> &amp; <font color="Maroon" title="c3">P3</font> <a href="topreal1.html#R1" title="TOPREAL1:pred.1">is_an_arc_of</a> <font color="Maroon" title="c4">p1</font>,<font color="Maroon" title="c5">p2</font> &amp; <font color="Maroon" title="c2">P2</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c3">P3</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"><font color="Maroon" title="c4">p1</font>,<font color="Maroon" title="c5">p2</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> &amp; <font color="Maroon" title="c1">P1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">P2</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c3">P3</font> &amp; not <font color="Maroon" title="c1">P1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">P2</font> implies <font color="Maroon" title="c1">P1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">P3</font> )</span><br/>



<b>assume </b><b>that </b><br/><a NAME="E1:12"/><i><font color="Green" title="E8">A1</font></i>: 
<font color="Maroon" title="c1">P1</font> <a href="topreal1.html#R1" title="TOPREAL1:pred.1">is_an_arc_of</a> <font color="Maroon" title="c4">p1</font>,<font color="Maroon" title="c5">p2</font>
 <b>and </b><br/><a NAME="E2:12"/><i><font color="Green" title="E9">A2</font></i>: 
<font color="Maroon" title="c2">P2</font> <a href="topreal1.html#R1" title="TOPREAL1:pred.1">is_an_arc_of</a> <font color="Maroon" title="c4">p1</font>,<font color="Maroon" title="c5">p2</font>
 <b>and </b><br/><a NAME="E3:12"/><i><font color="Green" title="E10">A3</font></i>: 
<font color="Maroon" title="c3">P3</font> <a href="topreal1.html#R1" title="TOPREAL1:pred.1">is_an_arc_of</a> <font color="Maroon" title="c4">p1</font>,<font color="Maroon" title="c5">p2</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c2">P2</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c3">P3</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"><font color="Maroon" title="c4">p1</font>,<font color="Maroon" title="c5">p2</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> or  not <font color="Maroon" title="c1">P1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">P2</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c3">P3</font> or <font color="Maroon" title="c1">P1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">P2</font> or <font color="Maroon" title="c1">P1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">P3</font> )</span><br/>

<b>assume </b><b>that </b><br/><a NAME="E4:12"/><i><font color="Green" title="E11">A4</font></i>: 
<font color="Maroon" title="c2">P2</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c3">P3</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"><font color="Maroon" title="c4">p1</font>,<font color="Maroon" title="c5">p2</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>
 <b>and </b><br/><a NAME="E5:12"/><i><font color="Green" title="E12">A5</font></i>: 
<font color="Maroon" title="c1">P1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">P2</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c3">P3</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c1">P1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">P2</font> or <font color="Maroon" title="c1">P1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">P3</font> )</span><br/>

<b>set </b><font color="Maroon" title="c6">P</font> = <font color="Maroon" title="c2">P2</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c3">P3</font>;<br/>
<a NAME="E6:12"/>
<font color="Maroon" title="c4">p1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">P2</font>
 
<b>by </b><i><a class="txt" href="#E2:12"><i><font color="Green" title="E9">A2</font></i></a>, <a class="ref" href="topreal1.html#T4" title="TOPREAL1:th.4">TOPREAL1:4</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c7">Q</font> = <font color="Maroon" title="c2">P2</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c3">P3</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="E8:12"/><i><font color="Green" title="E13">A7</font></i>: 
<font color="Maroon" title="c5">p2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">P2</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c3">P3</font>
 
<b>by </b><i><a class="txt" href="#E4:12"><i><font color="Green" title="E11">A4</font></i></a>, <a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a></i>;<br/>
<a NAME="E9:12"/><i><font color="Green" title="E14">A8</font></i>: 
<font color="Maroon" title="c4">p1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">P2</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c3">P3</font>
 
<b>by </b><i><a class="txt" href="#E4:12"><i><font color="Green" title="E11">A4</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="c8">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="c1">P1</font></span>)</span><b> such that </b><br/><a NAME="E11:12"/><i><font color="Green" title="E15">A9</font></i>: 
( <font color="Maroon" title="c8">f</font> is <a href="tops_2.html#V3" title="TOPS_2:attr.3">being_homeomorphism</a> &amp; <font color="Maroon" title="c8">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> <font color="Maroon" title="c4">p1</font> &amp; <font color="Maroon" title="c8">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> <font color="Maroon" title="c5">p2</font> )
 <b>by </b><i><a class="txt" href="#E1:12"><i><font color="Green" title="E8">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="E16">A10</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> <span class="p2">(<span class="default"><font color="Maroon" title="c2">P2</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c3">P3</font></span>)</span></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> <span class="p2">(<span class="default"><font color="Maroon" title="c2">P2</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c3">P3</font></span>)</span></span>)</span>

<br/>.= 
<font color="Maroon" title="c2">P2</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c3">P3</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="E13:12"/><i><font color="Green" title="E17">A11</font></i>: 
<font color="Maroon" title="c8">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="#E11:12"><i><font color="Green" title="E15">A9</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="E14:12"/><i><font color="Green" title="E18">A12</font></i>: 
<font color="Maroon" title="c8">f</font> is <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a>
 
<b>by </b><i><a class="txt" href="#E11:12"><i><font color="Green" title="E15">A9</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="E15:12"/><i><font color="Green" title="E19">A13</font></i>: 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c8">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="#E11:12"><i><font color="Green" title="E15">A9</font></i></a>, <a class="ref" href="tops_2.html#D5" title="TOPS_2:def.5">TOPS_2:def 5</a></i>;<br/>
<i><font color="Green" title="E20">A14</font></i>:  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c8">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="c1">P1</font></span>)</span>
<b>by </b><i><a class="txt" href="#E11:12"><i><font color="Green" title="E15">A9</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="c1">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/>
<b>reconsider </b><font color="Maroon" title="c9">U2</font> = <font color="Maroon" title="c2">P2</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> <span class="p2">(<span class="default"><font color="Maroon" title="c2">P2</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c3">P3</font></span>)</span></span>)</span> <b>by </b><i><a class="txt" href="#E12:12"><i><font color="Green" title="E16">A10</font></i></a>, <a class="ref" href="xboole_1.html#T7" title="XBOOLE_1:th.7">XBOOLE_1:7</a></i>;<br/>
<a NAME="E18:12"/>
<font color="Maroon" title="c9">U2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">P2</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">P2</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c3">P3</font></span>)</span>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T7" title="XBOOLE_1:th.7">XBOOLE_1:7</a>, <a class="ref" href="xboole_1.html#T28" title="XBOOLE_1:th.28">XBOOLE_1:28</a></i>;<br/>
<a NAME="E19:12"/><b>then </b><i><font color="Green" title="E21">A15</font></i>: 
<font color="Maroon" title="c9">U2</font> is <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a>
 
<b>by </b><i><a class="txt" href="#E2:12"><i><font color="Green" title="E9">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="c10">U3</font> = <font color="Maroon" title="c3">P3</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> <span class="p2">(<span class="default"><font color="Maroon" title="c2">P2</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c3">P3</font></span>)</span></span>)</span> <b>by </b><i><a class="txt" href="#E12:12"><i><font color="Green" title="E16">A10</font></i></a>, <a class="ref" href="xboole_1.html#T7" title="XBOOLE_1:th.7">XBOOLE_1:7</a></i>;<br/>
<a NAME="E21:12"/>
<font color="Maroon" title="c10">U3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">P3</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">P2</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c3">P3</font></span>)</span>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T7" title="XBOOLE_1:th.7">XBOOLE_1:7</a>, <a class="ref" href="xboole_1.html#T28" title="XBOOLE_1:th.28">XBOOLE_1:28</a></i>;<br/>
<a NAME="E22:12"/><b>then </b><i><font color="Green" title="E22">A16</font></i>: 
<font color="Maroon" title="c10">U3</font> is <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a>
 
<b>by </b><i><a class="txt" href="#E3:12"><i><font color="Green" title="E10">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="hsNdiv(this)" href="javascript:()" title="12_3"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:12_3"/>
( 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#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">r</font> &amp; <font color="Olive" title="b1">r</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1 holds <br/><font color="Maroon" title="c8">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> <font color="Maroon" title="c3">P3</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#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">r</font> &amp; <font color="Olive" title="b1">r</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1 &amp; not <font color="Maroon" title="c8">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> <font color="Maroon" title="c3">P3</font> ) )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="12_3_1"><b>suppose </b></a><a NAME="E1:12_3_1"/><i><font color="Green" title="E23">A17</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#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">r</font> &amp; <font color="Olive" title="b1">r</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1 holds <br/><font color="Maroon" title="c8">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> <font color="Maroon" title="c3">P3</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c1">P1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">P2</font> or <font color="Maroon" title="c1">P1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">P3</font> )</span><br/><div class="add"><a NAME="E2:12_3_1"/>
<font color="Maroon" title="c1">P1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">P3</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_3_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c11">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> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c11">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">P1</font> or <font color="Maroon" title="c11">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">P3</font> )</span><br/>

<b>assume </b><a NAME="E1:12_3_1_1"/>
<font color="Maroon" title="c11">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">P1</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c11">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">P3</font></span><br/>

<b>then consider </b><font color="Maroon" title="c12">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:12_3_1_1"/><i><font color="Green" title="E24">A18</font></i>: 
( <font color="Maroon" title="c12">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="c8">f</font> &amp; <font color="Maroon" title="c11">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">x</font> )
 <b>by </b><i><a class="txt" href="#E16:12"><i><font color="Green" title="E20">A14</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="c13">r</font> = <font color="Maroon" title="c12">x</font> as   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> <b>by </b><i><a class="txt" href="#E15:12"><i><font color="Green" title="E19">A13</font></i></a>, <a class="txt" href="#E3:12_3_1_1"><i><font color="Green" title="E24">A18</font></i></a>, <a class="ref" href="borsuk_1.html#T83" title="BORSUK_1:th.83">BORSUK_1:83</a></i>;<br/>
<a NAME="E5:12_3_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="c13">r</font> &amp; <font color="Maroon" title="c13">r</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 )
 
<b>by </b><i><a class="txt" href="#E3:12_3_1_1"><i><font color="Green" title="E24">A18</font></i></a>, <a class="ref" href="borsuk_1.html#T83" title="BORSUK_1:th.83">BORSUK_1:83</a>, <a class="ref" href="xxreal_1.html#T1" title="XXREAL_1:th.1">XXREAL_1:1</a></i>;<br/>
<a NAME="E6:12_3_1_1"/><b>then </b>
( <font color="Maroon" title="c13">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="c13">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or (  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c13">r</font> &amp; <font color="Maroon" title="c13">r</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1 ) )
 
<b>by </b><i><a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></i>;<br/>
<b>hence </b><a NAME="E7:12_3_1_1"/>
<font color="Maroon" title="c11">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">P3</font>
 <b>by </b><i><a class="txt" href="#E8:12"><i><font color="Green" title="E13">A7</font></i></a>, <a class="txt" href="#E9:12"><i><font color="Green" title="E14">A8</font></i></a>, <a class="txt" href="#E11:12"><i><font color="Green" title="E15">A9</font></i></a>, <a class="txt" href="#E1:12_3_1"><i><font color="Green" title="E23">A17</font></i></a>, <a class="txt" href="#E3:12_3_1_1"><i><font color="Green" title="E24">A18</font></i></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div><b>hence </b><a NAME="E3:12_3_1"/>
( <font color="Maroon" title="c1">P1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">P2</font> or <font color="Maroon" title="c1">P1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">P3</font> )
 <b>by </b><i><a class="txt" href="#E1:12"><i><font color="Green" title="E8">A1</font></i></a>, <a class="txt" href="#E3:12"><i><font color="Green" title="E10">A3</font></i></a>, <a class="ref" href="jordan6.html#T59" title="JORDAN6:th.59">JORDAN6:59</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="12_3_2"><b>suppose </b></a><a NAME="E1:12_3_2"/><i><font color="Green" title="E23">A19</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#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">r</font> &amp; <font color="Olive" title="b1">r</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1 &amp; not <font color="Maroon" title="c8">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> <font color="Maroon" title="c3">P3</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c1">P1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">P2</font> or <font color="Maroon" title="c1">P1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">P3</font> )</span><br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_3_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="12_3_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:12_3_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#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">r</font> &amp; <font color="Olive" title="b1">r</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1 holds <br/><font color="Maroon" title="c8">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> <font color="Maroon" title="c2">P2</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#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">r</font> &amp; <font color="Olive" title="b1">r</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1 &amp; not <font color="Maroon" title="c8">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> <font color="Maroon" title="c2">P2</font> ) )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="12_3_2_1_1_1"><b>case </b></a><a NAME="E1:12_3_2_1_1_1"/><i><font color="Green" title="E24">A20</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#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">r</font> &amp; <font color="Olive" title="b1">r</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1 holds <br/><font color="Maroon" title="c8">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> <font color="Maroon" title="c2">P2</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c1">P1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">P2</font> or <font color="Maroon" title="c1">P1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">P3</font> )</span><br/><div class="add"><a NAME="E2:12_3_2_1_1_1"/>
<font color="Maroon" title="c1">P1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">P2</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_3_2_1_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c11">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> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c11">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">P1</font> or <font color="Maroon" title="c11">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">P2</font> )</span><br/>

<b>assume </b><a NAME="E1:12_3_2_1_1_1_1"/>
<font color="Maroon" title="c11">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">P1</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c11">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">P2</font></span><br/>

<b>then consider </b><font color="Maroon" title="c12">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:12_3_2_1_1_1_1"/><i><font color="Green" title="E25">A21</font></i>: 
( <font color="Maroon" title="c12">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="c8">f</font> &amp; <font color="Maroon" title="c11">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">x</font> )
 <b>by </b><i><a class="txt" href="#E16:12"><i><font color="Green" title="E20">A14</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="c13">r</font> = <font color="Maroon" title="c12">x</font> as   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> <b>by </b><i><a class="txt" href="#E15:12"><i><font color="Green" title="E19">A13</font></i></a>, <a class="txt" href="#E3:12_3_2_1_1_1_1"><i><font color="Green" title="E25">A21</font></i></a>, <a class="ref" href="borsuk_1.html#T83" title="BORSUK_1:th.83">BORSUK_1:83</a></i>;<br/>
<a NAME="E5:12_3_2_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="c13">r</font> &amp; <font color="Maroon" title="c13">r</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 )
 
<b>by </b><i><a class="txt" href="#E3:12_3_2_1_1_1_1"><i><font color="Green" title="E25">A21</font></i></a>, <a class="ref" href="borsuk_1.html#T83" title="BORSUK_1:th.83">BORSUK_1:83</a>, <a class="ref" href="xxreal_1.html#T1" title="XXREAL_1:th.1">XXREAL_1:1</a></i>;<br/>
<a NAME="E6:12_3_2_1_1_1_1"/><b>then </b>
( (  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c13">r</font> &amp; <font color="Maroon" title="c13">r</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1 ) or <font color="Maroon" title="c13">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="c13">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 )
 
<b>by </b><i><a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></i>;<br/>
<b>hence </b><a NAME="E7:12_3_2_1_1_1_1"/>
<font color="Maroon" title="c11">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">P2</font>
 <b>by </b><i><a class="txt" href="#E8:12"><i><font color="Green" title="E13">A7</font></i></a>, <a class="txt" href="#E9:12"><i><font color="Green" title="E14">A8</font></i></a>, <a class="txt" href="#E11:12"><i><font color="Green" title="E15">A9</font></i></a>, <a class="txt" href="#E1:12_3_2_1_1_1"><i><font color="Green" title="E24">A20</font></i></a>, <a class="txt" href="#E3:12_3_2_1_1_1_1"><i><font color="Green" title="E25">A21</font></i></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div><b>hence </b><a NAME="E3:12_3_2_1_1_1"/>
( <font color="Maroon" title="c1">P1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">P2</font> or <font color="Maroon" title="c1">P1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">P3</font> )
 <b>by </b><i><a class="txt" href="#E1:12"><i><font color="Green" title="E8">A1</font></i></a>, <a class="txt" href="#E2:12"><i><font color="Green" title="E9">A2</font></i></a>, <a class="ref" href="jordan6.html#T59" title="JORDAN6:th.59">JORDAN6:59</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="12_3_2_1_1_2"><b>case </b></a><a NAME="E1:12_3_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#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">r</font> &amp; <font color="Olive" title="b1">r</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1 &amp; not <font color="Maroon" title="c8">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> <font color="Maroon" title="c2">P2</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/><div class="add"><b>then consider </b><font color="Maroon" title="c11">r2</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a><b> such that </b><br/><a NAME="E3:12_3_2_1_1_2"/><i><font color="Green" title="E24">A22</font></i>: 
(  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c11">r2</font> &amp; <font color="Maroon" title="c11">r2</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1 &amp; not <font color="Maroon" title="c8">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">r2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">P2</font> )
 ;<br/><b>consider </b><font color="Maroon" title="c12">r1</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a><b> such that </b><br/><a NAME="E5:12_3_2_1_1_2"/><i><font color="Green" title="E25">A23</font></i>: 
(  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c12">r1</font> &amp; <font color="Maroon" title="c12">r1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1 &amp; not <font color="Maroon" title="c8">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">r1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">P3</font> )
 <b>by </b><i><a class="txt" href="#E1:12_3_2"><i><font color="Green" title="E23">A19</font></i></a></i>;<br/><a NAME="E6:12_3_2_1_1_2"/>
<font color="Maroon" title="c11">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="#E3:12_3_2_1_1_2"><i><font color="Green" title="E24">A22</font></i></a>, <a class="ref" href="xxreal_1.html#T1" title="XXREAL_1:th.1">XXREAL_1:1</a></i>;<br/><a NAME="E7:12_3_2_1_1_2"/><b>then </b>
<font color="Maroon" title="c8">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">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="c8">f</font>
 <b>by </b><i><a class="txt" href="#E15:12"><i><font color="Green" title="E19">A13</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="E8:12_3_2_1_1_2"/><b>then </b><i><font color="Green" title="E26">A24</font></i>: 
<font color="Maroon" title="c8">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">r2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">P3</font>
 <b>by </b><i><a class="txt" href="#E5:12"><i><font color="Green" title="E12">A5</font></i></a>, <a class="txt" href="#E16:12"><i><font color="Green" title="E20">A14</font></i></a>, <a class="txt" href="#E3:12_3_2_1_1_2"><i><font color="Green" title="E24">A22</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/><a NAME="E9:12_3_2_1_1_2"/>
<font color="Maroon" title="c12">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="#E5:12_3_2_1_1_2"><i><font color="Green" title="E25">A23</font></i></a>, <a class="ref" href="xxreal_1.html#T1" title="XXREAL_1:th.1">XXREAL_1:1</a></i>;<br/><a NAME="E10:12_3_2_1_1_2"/><b>then </b><i><font color="Green" title="E27">A25</font></i>: 
<font color="Maroon" title="c8">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">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="c8">f</font>
 <b>by </b><i><a class="txt" href="#E15:12"><i><font color="Green" title="E19">A13</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="E11:12_3_2_1_1_2"/><b>then </b><i><font color="Green" title="E28">A26</font></i>: 
<font color="Maroon" title="c8">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">r1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">P2</font>
 <b>by </b><i><a class="txt" href="#E5:12"><i><font color="Green" title="E12">A5</font></i></a>, <a class="txt" href="#E16:12"><i><font color="Green" title="E20">A14</font></i></a>, <a class="txt" href="#E5:12_3_2_1_1_2"><i><font color="Green" title="E25">A23</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_3_2_1_1_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="12_3_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="E1:12_3_2_1_1_2_1_1"/>
( <font color="Maroon" title="c12">r1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c11">r2</font> or <font color="Maroon" title="c12">r1</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c11">r2</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="12_3_2_1_1_2_1_1_1"><b>suppose </b></a><a NAME="E1:12_3_2_1_1_2_1_1_1"/><i><font color="Green" title="E29">A27</font></i>: 
<font color="Maroon" title="c12">r1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c11">r2</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_3_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="12_3_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="E1:12_3_2_1_1_2_1_1_1_1_1"/>
( <font color="Maroon" title="c12">r1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">r2</font> or <font color="Maroon" title="c12">r1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c11">r2</font> )
 </span><b>by </b><i><a class="txt" href="#E1:12_3_2_1_1_2_1_1_1"><i><font color="Green" title="E29">A27</font></i></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="12_3_2_1_1_2_1_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:12_3_2_1_1_2_1_1_1_1_1_1"/>
<font color="Maroon" title="c12">r1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">r2</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/><div class="add"><b>hence </b><a NAME="E2:12_3_2_1_1_2_1_1_1_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="#E5:12"><i><font color="Green" title="E12">A5</font></i></a>, <a class="txt" href="#E16:12"><i><font color="Green" title="E20">A14</font></i></a>, <a class="txt" href="#E3:12_3_2_1_1_2"><i><font color="Green" title="E24">A22</font></i></a>, <a class="txt" href="#E5:12_3_2_1_1_2"><i><font color="Green" title="E25">A23</font></i></a>, <a class="txt" href="#E10:12_3_2_1_1_2"><i><font color="Green" title="E27">A25</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="12_3_2_1_1_2_1_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:12_3_2_1_1_2_1_1_1_1_1_2"/><i><font color="Green" title="E30">A28</font></i>: 
<font color="Maroon" title="c12">r1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c11">r2</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/><div class="add"><i><font color="Green" title="E31">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> <span class="p2">(<span class="default"><font color="Maroon" title="c2">P2</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c3">P3</font></span>)</span></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> <span class="p2">(<span class="default"><font color="Maroon" title="c2">P2</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c3">P3</font></span>)</span></span>)</span>

<br/>.= 
<font color="Maroon" title="c2">P2</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c3">P3</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="c1">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="c1">P1</font></span>)</span>

<br/>.= 
<font color="Maroon" title="c1">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="E4:12_3_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="c8">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> <span class="p2">(<span class="default"><font color="Maroon" title="c2">P2</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c3">P3</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="#E5:12"><i><font color="Green" title="E12">A5</font></i></a>, <a class="txt" href="#E2:12_3_2_1_1_2_1_1_1_1_1_2"><i><font color="Green" title="E31">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="c13">g</font> = <font color="Maroon" title="c8">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> <span class="p2">(<span class="default"><font color="Maroon" title="c2">P2</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c3">P3</font></span>)</span></span>)</span> <b>by </b><i><a class="txt" href="#E15:12"><i><font color="Green" title="E19">A13</font></i></a>, <a class="ref" href="funct_2.html#T4" title="FUNCT_2:th.4">FUNCT_2:4</a></i>;<br/><a NAME="E6:12_3_2_1_1_2_1_1_1_1_1_2"/>
<font color="Maroon" title="c2">P2</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c3">P3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">P1</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">P2</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c3">P3</font></span>)</span>
 <b>by </b><i><a class="txt" href="#E5:12"><i><font color="Green" title="E12">A5</font></i></a>, <a class="ref" href="xboole_1.html#T12" title="XBOOLE_1:th.12">XBOOLE_1:12</a></i>;<br/><a NAME="E7:12_3_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="c1">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> <span class="p1">(<span class="default"><font color="Maroon" title="c2">P2</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c3">P3</font></span>)</span>
 <b>by </b><i><a class="ref" href="topmetr.html#T5" title="TOPMETR:th.5">TOPMETR:5</a></i>;<br/><a NAME="E8:12_3_2_1_1_2_1_1_1_1_1_2"/><b>then </b><i><font color="Green" title="E32">A30</font></i>: 
<font color="Maroon" title="c13">g</font> is <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a>
 <b>by </b><i><a class="txt" href="#E14:12"><i><font color="Green" title="E18">A12</font></i></a>, <a class="ref" href="pre_topc.html#T56" title="PRE_TOPC:th.56">PRE_TOPC:56</a></i>;<br/><a NAME="E9:12_3_2_1_1_2_1_1_1_1_1_2"/><i><font color="Green" title="E33">A31</font></i>: 
<font color="Maroon" title="c9">U2</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c10">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="c7">Q</font></span>)</span>
 <b>by </b><i><a class="ref" href="topmetr.html#D2" title="TOPMETR:def.2">TOPMETR:def 2</a></i>;<br/><a NAME="E10:12_3_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> <span class="p1">(<span class="default"><font color="Maroon" title="c2">P2</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c3">P3</font></span>)</span> <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="c7">Q</font></span>)</span>
 <b>by </b><i><a class="ref" href="euclid.html#T67" title="EUCLID:th.67">EUCLID:67</a></i>;<br/><b>then consider </b><font color="Maroon" title="c14">r0</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a><b> such that </b><br/><a NAME="E12:12_3_2_1_1_2_1_1_1_1_1_2"/><i><font color="Green" title="E34">A32</font></i>: 
( <font color="Maroon" title="c12">r1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c14">r0</font> &amp; <font color="Maroon" title="c14">r0</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c11">r2</font> &amp; <font color="Maroon" title="c13">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">r0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">U2</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c10">U3</font> )
 <b>by </b><i><a class="txt" href="#E19:12"><i><font color="Green" title="E21">A15</font></i></a>, <a class="txt" href="#E22:12"><i><font color="Green" title="E22">A16</font></i></a>, <a class="txt" href="#E3:12_3_2_1_1_2"><i><font color="Green" title="E24">A22</font></i></a>, <a class="txt" href="#E5:12_3_2_1_1_2"><i><font color="Green" title="E25">A23</font></i></a>, <a class="txt" href="#E8:12_3_2_1_1_2"><i><font color="Green" title="E26">A24</font></i></a>, <a class="txt" href="#E11:12_3_2_1_1_2"><i><font color="Green" title="E28">A26</font></i></a>, <a class="txt" href="#E1:12_3_2_1_1_2_1_1_1_1_1_2"><i><font color="Green" title="E30">A28</font></i></a>, <a class="txt" href="#E8:12_3_2_1_1_2_1_1_1_1_1_2"><i><font color="Green" title="E32">A30</font></i></a>, <a class="txt" href="#E9:12_3_2_1_1_2_1_1_1_1_1_2"><i><font color="Green" title="E33">A31</font></i></a>, <a class="ref" href="topmetr3.html#T17" title="TOPMETR3:th.17">TOPMETR3:17</a></i>;<br/><a NAME="E13:12_3_2_1_1_2_1_1_1_1_1_2"/><i><font color="Green" title="E35">A33</font></i>: 
( <font color="Maroon" title="c13">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">r0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">p1</font> or <font color="Maroon" title="c13">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">r0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">p2</font> )
 <b>by </b><i><a class="txt" href="#E4:12"><i><font color="Green" title="E11">A4</font></i></a>, <a class="txt" href="#E12:12_3_2_1_1_2_1_1_1_1_1_2"><i><font color="Green" title="E34">A32</font></i></a>, <a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a></i>;<br/><a NAME="E14:12_3_2_1_1_2_1_1_1_1_1_2"/>
<font color="Maroon" title="c14">r0</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1
 <b>by </b><i><a class="txt" href="#E3:12_3_2_1_1_2"><i><font color="Green" title="E24">A22</font></i></a>, <a class="txt" href="#E12:12_3_2_1_1_2_1_1_1_1_1_2"><i><font color="Green" title="E34">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="E15:12_3_2_1_1_2_1_1_1_1_1_2"/><b>then </b><i><font color="Green" title="E36">A34</font></i>: 
<font color="Maroon" title="c14">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="c8">f</font>
 <b>by </b><i><a class="txt" href="#E15:12"><i><font color="Green" title="E19">A13</font></i></a>, <a class="txt" href="#E5:12_3_2_1_1_2"><i><font color="Green" title="E25">A23</font></i></a>, <a class="txt" href="#E12:12_3_2_1_1_2_1_1_1_1_1_2"><i><font color="Green" title="E34">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="xxreal_1.html#T1" title="XXREAL_1:th.1">XXREAL_1:1</a></i>;<br/><a NAME="E16:12_3_2_1_1_2_1_1_1_1_1_2"/><i><font color="Green" title="E37">A35</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="c8">f</font>
 <b>by </b><i><a class="txt" href="#E15:12"><i><font color="Green" title="E19">A13</font></i></a>, <a class="ref" href="borsuk_1.html#T83" title="BORSUK_1:th.83">BORSUK_1:83</a>, <a class="ref" href="xxreal_1.html#T1" title="XXREAL_1:th.1">XXREAL_1:1</a></i>;<br/><a NAME="E17:12_3_2_1_1_2_1_1_1_1_1_2"/>
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="c8">f</font>
 <b>by </b><i><a class="txt" href="#E15:12"><i><font color="Green" title="E19">A13</font></i></a>, <a class="ref" href="borsuk_1.html#T83" title="BORSUK_1:th.83">BORSUK_1:83</a>, <a class="ref" href="xxreal_1.html#T1" title="XXREAL_1:th.1">XXREAL_1:1</a></i>;<br/><b>hence </b><a NAME="E18:12_3_2_1_1_2_1_1_1_1_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="#E11:12"><i><font color="Green" title="E15">A9</font></i></a>, <a class="txt" href="#E13:12"><i><font color="Green" title="E17">A11</font></i></a>, <a class="txt" href="#E3:12_3_2_1_1_2"><i><font color="Green" title="E24">A22</font></i></a>, <a class="txt" href="#E5:12_3_2_1_1_2"><i><font color="Green" title="E25">A23</font></i></a>, <a class="txt" href="#E12:12_3_2_1_1_2_1_1_1_1_1_2"><i><font color="Green" title="E34">A32</font></i></a>, <a class="txt" href="#E13:12_3_2_1_1_2_1_1_1_1_1_2"><i><font color="Green" title="E35">A33</font></i></a>, <a class="txt" href="#E15:12_3_2_1_1_2_1_1_1_1_1_2"><i><font color="Green" title="E36">A34</font></i></a>, <a class="txt" href="#E16:12_3_2_1_1_2_1_1_1_1_1_2"><i><font color="Green" title="E37">A35</font></i></a>, <a class="ref" href="funct_1.html#D8" title="FUNCT_1:def.8">FUNCT_1:def 8</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E3:12_3_2_1_1_2_1_1_1"/>
contradiction
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="12_3_2_1_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:12_3_2_1_1_2_1_1_2"/><i><font color="Green" title="E29">A36</font></i>: 
<font color="Maroon" title="c12">r1</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c11">r2</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/><div class="add"><i><font color="Green" title="E30">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> <span class="p2">(<span class="default"><font color="Maroon" title="c2">P2</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c3">P3</font></span>)</span></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> <span class="p2">(<span class="default"><font color="Maroon" title="c2">P2</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c3">P3</font></span>)</span></span>)</span>

<br/>.= 
<font color="Maroon" title="c2">P2</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c3">P3</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="c1">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="c1">P1</font></span>)</span>

<br/>.= 
<font color="Maroon" title="c1">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="E4:12_3_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="c8">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> <span class="p2">(<span class="default"><font color="Maroon" title="c2">P2</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c3">P3</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="#E5:12"><i><font color="Green" title="E12">A5</font></i></a>, <a class="txt" href="#E2:12_3_2_1_1_2_1_1_2"><i><font color="Green" title="E30">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="c13">g</font> = <font color="Maroon" title="c8">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> <span class="p2">(<span class="default"><font color="Maroon" title="c2">P2</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c3">P3</font></span>)</span></span>)</span> <b>by </b><i><a class="txt" href="#E15:12"><i><font color="Green" title="E19">A13</font></i></a>, <a class="ref" href="funct_2.html#T4" title="FUNCT_2:th.4">FUNCT_2:4</a></i>;<br/><a NAME="E6:12_3_2_1_1_2_1_1_2"/>
<font color="Maroon" title="c2">P2</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c3">P3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">P1</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">P2</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c3">P3</font></span>)</span>
 <b>by </b><i><a class="txt" href="#E5:12"><i><font color="Green" title="E12">A5</font></i></a>, <a class="ref" href="xboole_1.html#T12" title="XBOOLE_1:th.12">XBOOLE_1:12</a></i>;<br/><a NAME="E7:12_3_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="c1">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> <span class="p1">(<span class="default"><font color="Maroon" title="c2">P2</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c3">P3</font></span>)</span>
 <b>by </b><i><a class="ref" href="topmetr.html#T5" title="TOPMETR:th.5">TOPMETR:5</a></i>;<br/><a NAME="E8:12_3_2_1_1_2_1_1_2"/><b>then </b><i><font color="Green" title="E31">A38</font></i>: 
<font color="Maroon" title="c13">g</font> is <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a>
 <b>by </b><i><a class="txt" href="#E14:12"><i><font color="Green" title="E18">A12</font></i></a>, <a class="ref" href="pre_topc.html#T56" title="PRE_TOPC:th.56">PRE_TOPC:56</a></i>;<br/><a NAME="E9:12_3_2_1_1_2_1_1_2"/><i><font color="Green" title="E32">A39</font></i>: 
<font color="Maroon" title="c9">U2</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c10">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="c7">Q</font></span>)</span>
 <b>by </b><i><a class="ref" href="topmetr.html#D2" title="TOPMETR:def.2">TOPMETR:def 2</a></i>;<br/><a NAME="E10:12_3_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> <span class="p1">(<span class="default"><font color="Maroon" title="c2">P2</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c3">P3</font></span>)</span> <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="c7">Q</font></span>)</span>
 <b>by </b><i><a class="ref" href="euclid.html#T67" title="EUCLID:th.67">EUCLID:67</a></i>;<br/><b>then consider </b><font color="Maroon" title="c14">r0</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a><b> such that </b><br/><a NAME="E12:12_3_2_1_1_2_1_1_2"/><i><font color="Green" title="E33">A40</font></i>: 
( <font color="Maroon" title="c11">r2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c14">r0</font> &amp; <font color="Maroon" title="c14">r0</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c12">r1</font> &amp; <font color="Maroon" title="c13">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">r0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">U2</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c10">U3</font> )
 <b>by </b><i><a class="txt" href="#E19:12"><i><font color="Green" title="E21">A15</font></i></a>, <a class="txt" href="#E22:12"><i><font color="Green" title="E22">A16</font></i></a>, <a class="txt" href="#E3:12_3_2_1_1_2"><i><font color="Green" title="E24">A22</font></i></a>, <a class="txt" href="#E5:12_3_2_1_1_2"><i><font color="Green" title="E25">A23</font></i></a>, <a class="txt" href="#E8:12_3_2_1_1_2"><i><font color="Green" title="E26">A24</font></i></a>, <a class="txt" href="#E11:12_3_2_1_1_2"><i><font color="Green" title="E28">A26</font></i></a>, <a class="txt" href="#E1:12_3_2_1_1_2_1_1_2"><i><font color="Green" title="E29">A36</font></i></a>, <a class="txt" href="#E8:12_3_2_1_1_2_1_1_2"><i><font color="Green" title="E31">A38</font></i></a>, <a class="txt" href="#E9:12_3_2_1_1_2_1_1_2"><i><font color="Green" title="E32">A39</font></i></a>, <a class="ref" href="topmetr3.html#T17" title="TOPMETR3:th.17">TOPMETR3:17</a></i>;<br/><a NAME="E13:12_3_2_1_1_2_1_1_2"/><i><font color="Green" title="E34">A41</font></i>: 
( <font color="Maroon" title="c13">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">r0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">p1</font> or <font color="Maroon" title="c13">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">r0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">p2</font> )
 <b>by </b><i><a class="txt" href="#E4:12"><i><font color="Green" title="E11">A4</font></i></a>, <a class="txt" href="#E12:12_3_2_1_1_2_1_1_2"><i><font color="Green" title="E33">A40</font></i></a>, <a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a></i>;<br/><a NAME="E14:12_3_2_1_1_2_1_1_2"/>
<font color="Maroon" title="c14">r0</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1
 <b>by </b><i><a class="txt" href="#E5:12_3_2_1_1_2"><i><font color="Green" title="E25">A23</font></i></a>, <a class="txt" href="#E12:12_3_2_1_1_2_1_1_2"><i><font color="Green" title="E33">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="E15:12_3_2_1_1_2_1_1_2"/><b>then </b><i><font color="Green" title="E35">A42</font></i>: 
<font color="Maroon" title="c14">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="c8">f</font>
 <b>by </b><i><a class="txt" href="#E15:12"><i><font color="Green" title="E19">A13</font></i></a>, <a class="txt" href="#E3:12_3_2_1_1_2"><i><font color="Green" title="E24">A22</font></i></a>, <a class="txt" href="#E12:12_3_2_1_1_2_1_1_2"><i><font color="Green" title="E33">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="xxreal_1.html#T1" title="XXREAL_1:th.1">XXREAL_1:1</a></i>;<br/><a NAME="E16:12_3_2_1_1_2_1_1_2"/>
(  <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="c8">f</font> &amp; 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="c8">f</font> )
 <b>by </b><i><a class="txt" href="#E15:12"><i><font color="Green" title="E19">A13</font></i></a>, <a class="ref" href="borsuk_1.html#T83" title="BORSUK_1:th.83">BORSUK_1:83</a>, <a class="ref" href="xxreal_1.html#T1" title="XXREAL_1:th.1">XXREAL_1:1</a></i>;<br/><b>hence </b><a NAME="E17:12_3_2_1_1_2_1_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="#E11:12"><i><font color="Green" title="E15">A9</font></i></a>, <a class="txt" href="#E13:12"><i><font color="Green" title="E17">A11</font></i></a>, <a class="txt" href="#E3:12_3_2_1_1_2"><i><font color="Green" title="E24">A22</font></i></a>, <a class="txt" href="#E5:12_3_2_1_1_2"><i><font color="Green" title="E25">A23</font></i></a>, <a class="txt" href="#E12:12_3_2_1_1_2_1_1_2"><i><font color="Green" title="E33">A40</font></i></a>, <a class="txt" href="#E13:12_3_2_1_1_2_1_1_2"><i><font color="Green" title="E34">A41</font></i></a>, <a class="txt" href="#E15:12_3_2_1_1_2_1_1_2"><i><font color="Green" title="E35">A42</font></i></a>, <a class="ref" href="funct_1.html#D8" title="FUNCT_1:def.8">FUNCT_1:def 8</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E13:12_3_2_1_1_2"/>
contradiction
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E3:12_3_2"/>
( <font color="Maroon" title="c1">P1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">P2</font> or <font color="Maroon" title="c1">P1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">P3</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div></div><b>end;</b></div>

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