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<span class="kw">let </span><font color="Maroon" title="c1">P</font> be  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>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">p1</font>, <font color="Olive" title="b2">p2</font>, <font color="Olive" title="b3">p</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><br/> for <font color="Olive" title="b4">e</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>  st <font color="Maroon" title="c1">P</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="Olive" title="b2">p2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Olive" title="b4">e</font> &amp; <font color="Olive" title="b3">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">P</font> &amp; <font color="Olive" title="b3">p</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">e</font> &amp; not <font color="Olive" title="b3">p</font> <a href="jordan20.html#R3" title="JORDAN20:pred.3">is_Lout</a> <font color="Maroon" title="c1">P</font>,<font color="Olive" title="b1">p1</font>,<font color="Olive" title="b2">p2</font>,<font color="Olive" title="b4">e</font> &amp; not <font color="Olive" title="b3">p</font> <a href="jordan20.html#R4" title="JORDAN20:pred.4">is_Rout</a> <font color="Maroon" title="c1">P</font>,<font color="Olive" title="b1">p1</font>,<font color="Olive" title="b2">p2</font>,<font color="Olive" title="b4">e</font> holds <br/><font color="Olive" title="b3">p</font> <a href="jordan20.html#R6" title="JORDAN20:pred.6">is_OSout</a> <font color="Maroon" title="c1">P</font>,<font color="Olive" title="b1">p1</font>,<font color="Olive" title="b2">p2</font>,<font color="Olive" title="b4">e</font></span><br/><span class="kw">let </span><font color="Maroon" title="c2">p1</font>, <font color="Maroon" title="c3">p2</font>, <font color="Maroon" title="c4">p</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">e</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>  st <font color="Maroon" title="c1">P</font> <a href="topreal1.html#R1" title="TOPREAL1:pred.1">is_an_arc_of</a> <font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font> &amp; <font color="Maroon" title="c3">p2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Olive" title="b1">e</font> &amp; <font color="Maroon" title="c4">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">P</font> &amp; <font color="Maroon" title="c4">p</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">e</font> &amp; not <font color="Maroon" title="c4">p</font> <a href="jordan20.html#R3" title="JORDAN20:pred.3">is_Lout</a> <font color="Maroon" title="c1">P</font>,<font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font>,<font color="Olive" title="b1">e</font> &amp; not <font color="Maroon" title="c4">p</font> <a href="jordan20.html#R4" title="JORDAN20:pred.4">is_Rout</a> <font color="Maroon" title="c1">P</font>,<font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font>,<font color="Olive" title="b1">e</font> holds <br/><font color="Maroon" title="c4">p</font> <a href="jordan20.html#R6" title="JORDAN20:pred.6">is_OSout</a> <font color="Maroon" title="c1">P</font>,<font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font>,<font color="Olive" title="b1">e</font></span><br/><span class="kw">let </span><font color="Maroon" title="c5">e</font> be   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c1">P</font> <a href="topreal1.html#R1" title="TOPREAL1:pred.1">is_an_arc_of</a> <font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font> &amp; <font color="Maroon" title="c3">p2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c5">e</font> &amp; <font color="Maroon" title="c4">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">P</font> &amp; <font color="Maroon" title="c4">p</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">e</font> &amp; not <font color="Maroon" title="c4">p</font> <a href="jordan20.html#R3" title="JORDAN20:pred.3">is_Lout</a> <font color="Maroon" title="c1">P</font>,<font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font>,<font color="Maroon" title="c5">e</font> &amp; not <font color="Maroon" title="c4">p</font> <a href="jordan20.html#R4" title="JORDAN20:pred.4">is_Rout</a> <font color="Maroon" title="c1">P</font>,<font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font>,<font color="Maroon" title="c5">e</font> implies <font color="Maroon" title="c4">p</font> <a href="jordan20.html#R6" title="JORDAN20:pred.6">is_OSout</a> <font color="Maroon" title="c1">P</font>,<font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font>,<font color="Maroon" title="c5">e</font> )</span><br/>





<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:11"/><span class="lab"><font color="Green" title="E12">A1</font></span>: 
<font color="Maroon" title="c1">P</font> <a href="topreal1.html#R1" title="TOPREAL1:pred.1">is_an_arc_of</a> <font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font>
 <span class="kw">and </span><br/><a NAME="E2:11"/><span class="lab"><font color="Green" title="E13">A2</font></span>: 
<font color="Maroon" title="c3">p2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c5">e</font>
 <span class="kw">and </span><br/><a NAME="E3:11"/><span class="lab"><font color="Green" title="E14">A3</font></span>: 
<font color="Maroon" title="c4">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">P</font>
 <span class="kw">and </span><br/><a NAME="E4:11"/><span class="lab"><font color="Green" title="E15">A4</font></span>: 
<font color="Maroon" title="c4">p</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">e</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c4">p</font> <a href="jordan20.html#R3" title="JORDAN20:pred.3">is_Lout</a> <font color="Maroon" title="c1">P</font>,<font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font>,<font color="Maroon" title="c5">e</font> or <font color="Maroon" title="c4">p</font> <a href="jordan20.html#R4" title="JORDAN20:pred.4">is_Rout</a> <font color="Maroon" title="c1">P</font>,<font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font>,<font color="Maroon" title="c5">e</font> or <font color="Maroon" title="c4">p</font> <a href="jordan20.html#R6" title="JORDAN20:pred.6">is_OSout</a> <font color="Maroon" title="c1">P</font>,<font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font>,<font color="Maroon" title="c5">e</font> )</span><br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="11_1"><span class="kw">now </span></a><div class="add"><span class="kw">reconsider </span><font color="Maroon" title="c6">pr1a</font> =  <a href="pscomp_1.html#K4" title="PSCOMP_1:func.4">proj1</a>  as   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</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="topmetr.html#K3" title="TOPMETR:func.3">R^1</a>  <span class="kw">by</span> <span class="lab"><a class="ref" href="topmetr.html#T24" title="TOPMETR:th.24">TOPMETR:24</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c7">pro1</font> = <font color="Maroon" title="c6">pr1a</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c1">P</font> as   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</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="topmetr.html#K3" title="TOPMETR:func.3">R^1</a>  <span class="kw">by</span> <span class="lab"><a class="ref" href="pre_topc.html#T30" title="PRE_TOPC:th.30">PRE_TOPC:30</a></span>;<br/><span class="kw">consider </span><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">P</font></span>)</span><span class="kw"> such that </span><br/><a NAME="E4:11_1"/><span class="lab"><font color="Green" title="E16">A5</font></span>: 
<font color="Maroon" title="c8">f</font> is  <a href="tops_2.html#V3" title="TOPS_2:attr.3">being_homeomorphism</a> 
 <span class="kw">and </span><br/><a NAME="E5:11_1"/><span class="lab"><font color="Green" title="E17">A6</font></span>: 
<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="c2">p1</font>
 <span class="kw">and </span><br/><a NAME="E6:11_1"/><span class="lab"><font color="Green" title="E18">A7</font></span>: 
<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="c3">p2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:11"><span class="lab"><font color="Green" title="E12">A1</font></span></a>, <a class="ref" href="topreal1.html#D2" title="TOPREAL1:def.2">TOPREAL1:def 2</a></span>;<br/><a NAME="E7:11_1"/><span class="lab"><font color="Green" title="E19">A8</font></span>: 
<font color="Maroon" title="c8">f</font> is  <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:11_1"><span class="lab"><font color="Green" title="E16">A5</font></span></a>, <a class="ref" href="tops_2.html#D5" title="TOPS_2:def.5">TOPS_2:def 5</a></span>;<br/><a NAME="E8:11_1"/><span class="lab"><font color="Green" title="E20">A9</font></span>: 
 <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</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> <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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:11_1"><span class="lab"><font color="Green" title="E16">A5</font></span></a>, <a class="ref" href="tops_2.html#D5" title="TOPS_2:def.5">TOPS_2:def 5</a></span>;<br/><a NAME="E9:11_1"/><span class="kw">then </span>
<font color="Maroon" title="c4">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c8">f</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:11"><span class="lab"><font color="Green" title="E14">A3</font></span></a>, <a class="ref" href="pre_topc.html#D10" title="PRE_TOPC:def.10">PRE_TOPC:def 10</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c9">xs</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E11:11_1"/><span class="lab"><font color="Green" title="E21">A10</font></span>: 
<font color="Maroon" title="c9">xs</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c8">f</font>
 <span class="kw">and </span><br/><a NAME="E12:11_1"/><span class="lab"><font color="Green" title="E22">A11</font></span>: 
<font color="Maroon" title="c4">p</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="c9">xs</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></span>;<br/><a NAME="E13:11_1"/><span class="lab"><font color="Green" title="E23">A12</font></span>: 
 <a href="relset_1.html#K1" title="RELSET_1:func.1">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> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:11_1"><span class="lab"><font color="Green" title="E16">A5</font></span></a>, <a class="ref" href="tops_2.html#D5" title="TOPS_2:def.5">TOPS_2:def 5</a></span>;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c10">s2</font> = <font color="Maroon" title="c9">xs</font> as   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:11_1"><span class="lab"><font color="Green" title="E21">A10</font></span></a>, <a class="ref" href="borsuk_1.html#T83" title="BORSUK_1:th.83">BORSUK_1:83</a></span>;<br/><a NAME="E15:11_1"/><span class="lab"><font color="Green" title="E24">A13</font></span>: 
<font color="Maroon" title="c10">s2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:11_1"><span class="lab"><font color="Green" title="E21">A10</font></span></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></span>;<br/><a NAME="E16:11_1"/>
for <font color="Olive" title="b1">q</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="Olive" title="b1">q</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> 1 holds <br/><font color="Olive" title="b1">q</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">e</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:11"><span class="lab"><font color="Green" title="E13">A2</font></span></a>, <a class="txt" href="#E6:11_1"><span class="lab"><font color="Green" title="E18">A7</font></span></a></span>;<br/><a NAME="E17:11_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E25">A14</font></span>: 
1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">s</font> where <font color="Olive" title="b1">s</font> is   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> : ( 1 <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Olive" title="b1">s</font> &amp; <font color="Olive" title="b1">s</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c10">s2</font> &amp; ( for <font color="Olive" title="b2">q</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="Olive" title="b2">q</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="Olive" title="b1">s</font> holds <br/><font color="Olive" title="b2">q</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">e</font> ) ) </span>}</span> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:11_1"><span class="lab"><font color="Green" title="E24">A13</font></span></a></span>;<br/><a NAME="E18:11_1"/>
<span class="p1">{<span class="default"> <font color="Olive" title="b1">s</font> where <font color="Olive" title="b1">s</font> is   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> : ( 1 <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Olive" title="b1">s</font> &amp; <font color="Olive" title="b1">s</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c10">s2</font> &amp; ( for <font color="Olive" title="b2">q</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="Olive" title="b2">q</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="Olive" title="b1">s</font> holds <br/><font color="Olive" title="b2">q</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">e</font> ) ) </span>}</span>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="11_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c11">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c11">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">s</font> where <font color="Olive" title="b1">s</font> is   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> : ( 1 <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Olive" title="b1">s</font> &amp; <font color="Olive" title="b1">s</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c10">s2</font> &amp; ( for <font color="Olive" title="b2">q</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="Olive" title="b2">q</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="Olive" title="b1">s</font> holds <br/><font color="Olive" title="b2">q</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">e</font> ) ) </span>}</span>  or <font color="Maroon" title="c11">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  )</span><br/>

<span class="kw">assume </span><a NAME="E1:11_1_1"/>
<font color="Maroon" title="c11">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">s</font> where <font color="Olive" title="b1">s</font> is   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> : ( 1 <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Olive" title="b1">s</font> &amp; <font color="Olive" title="b1">s</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c10">s2</font> &amp; ( for <font color="Olive" title="b2">q</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="Olive" title="b2">q</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="Olive" title="b1">s</font> holds <br/><font color="Olive" title="b2">q</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">e</font> ) ) </span>}</span> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c11">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> </span><br/>

<a NAME="E2:11_1_1"/><span class="kw">then </span>
 ex <font color="Olive" title="b1">s</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> st <br/>( <font color="Olive" title="b1">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">x</font> &amp; 1 <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Olive" title="b1">s</font> &amp; <font color="Olive" title="b1">s</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c10">s2</font> &amp; ( for <font color="Olive" title="b2">q</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="Olive" title="b2">q</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="Olive" title="b1">s</font> holds <br/><font color="Olive" title="b2">q</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">e</font> ) )
 
;<br/>
<span class="kw">hence </span><a NAME="E3:11_1_1"/>
<font color="Maroon" title="c11">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c11">R</font> = <span class="p1">{<span class="default"> <font color="Olive" title="b1">s</font> where <font color="Olive" title="b1">s</font> is   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> : ( 1 <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Olive" title="b1">s</font> &amp; <font color="Olive" title="b1">s</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c10">s2</font> &amp; ( for <font color="Olive" title="b2">q</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="Olive" title="b2">q</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="Olive" title="b1">s</font> holds <br/><font color="Olive" title="b2">q</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">e</font> ) ) </span>}</span>  as  non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  <span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:11_1"><span class="lab"><font color="Green" title="E25">A14</font></span></a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c12">s0</font> =  <a href="seq_4.html#K5" title="SEQ_4:func.5">lower_bound</a> <font color="Maroon" title="c11">R</font> as   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> ;<br/><a NAME="E21:11_1"/><span class="lab"><font color="Green" title="E26">A15</font></span>: 
for <font color="Olive" title="b1">s</font> being  <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>   <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>   st <font color="Olive" title="b1">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">R</font> holds <br/><font color="Olive" title="b1">s</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c10">s2</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="11_1_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c13">s</font> be   <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>   <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c13">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">R</font> implies <font color="Maroon" title="c13">s</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c10">s2</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:11_1_2"/>
<font color="Maroon" title="c13">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">R</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c13">s</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c10">s2</font></span><br/>

<a NAME="E2:11_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E27">A16</font></span>: 
 ex <font color="Olive" title="b1">s3</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> st <br/>( <font color="Olive" title="b1">s3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">s</font> &amp; 1 <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Olive" title="b1">s3</font> &amp; <font color="Olive" title="b1">s3</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c10">s2</font> &amp; ( for <font color="Olive" title="b2">q</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="Olive" title="b2">q</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="Olive" title="b1">s3</font> holds <br/><font color="Olive" title="b2">q</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">e</font> ) )
 
;<br/>
<a NAME="E3:11_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c13">s</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c10">s2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:11"><span class="lab"><font color="Green" title="E15">A4</font></span></a>, <a class="txt" href="#E12:11_1"><span class="lab"><font color="Green" title="E22">A11</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E4:11_1_2"/>
<font color="Maroon" title="c13">s</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c10">s2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:11_1_2"><span class="lab"><font color="Green" title="E27">A16</font></span></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><a NAME="E22:11_1"/><span class="kw">then </span>
for <font color="Olive" title="b1">s</font> being  <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>   <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>   st <font color="Olive" title="b1">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">R</font> holds <br/><font color="Olive" title="b1">s</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c10">s2</font>
 ;<br/><a NAME="E23:11_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E27">A17</font></span>: 
<font color="Maroon" title="c12">s0</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c10">s2</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="seq_4.html#T60" title="SEQ_4:th.60">SEQ_4:60</a></span>;<br/><a NAME="E24:11_1"/>
for <font color="Olive" title="b1">p7</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</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> holds  <font color="Maroon" title="c7">pro1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">p7</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="pscomp_1.html#K4" title="PSCOMP_1:func.4">proj1</a>  <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">p7</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="11_1_3"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c13">p7</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</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 class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c7">pro1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">p7</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="pscomp_1.html#K4" title="PSCOMP_1:func.4">proj1</a>  <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">p7</font></span><br/>

<a NAME="E1:11_1_3"/>
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="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">P</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="pre_topc.html#T29" title="PRE_TOPC:th.29">PRE_TOPC:29</a></span>;<br/>
<span class="kw">hence </span><a NAME="E2:11_1_3"/>
<font color="Maroon" title="c7">pro1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">p7</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="pscomp_1.html#K4" title="PSCOMP_1:func.4">proj1</a>  <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">p7</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#T72" title="FUNCT_1:th.72">FUNCT_1:72</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><a NAME="E25:11_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E28">A18</font></span>: 
<font color="Maroon" title="c7">pro1</font> is  <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="jgraph_2.html#T39" title="JGRAPH_2:th.39">JGRAPH_2:39</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c13">h</font> = <font color="Maroon" title="c7">pro1</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <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> ,<a href="topmetr.html#K3" title="TOPMETR:func.3">R^1</a>  ;<br/><a NAME="E27:11_1"/><span class="lab"><font color="Green" title="E29">A19</font></span>: 
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c13">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1</span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="borsuk_1.html#T83" title="BORSUK_1:th.83">BORSUK_1:83</a>, <a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></span>;<br/><a NAME="E28:11_1"/>
for <font color="Olive" title="b1">s</font> being  <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>   <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>   st <font color="Olive" title="b1">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">R</font> holds <br/><font color="Olive" title="b1">s</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c10">s2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E21:11_1"><span class="lab"><font color="Green" title="E26">A15</font></span></a></span>;<br/><a NAME="E29:11_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E30">A20</font></span>: 
<font color="Maroon" title="c11">R</font> is  <a href="xxreal_2.html#V3" title="XXREAL_2:attr.3">bounded_below</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="seq_4.html#D2" title="SEQ_4:def.2">SEQ_4:def 2</a></span>;<br/><a NAME="E30:11_1"/><span class="lab"><font color="Green" title="E31">A21</font></span>: 
 <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="c10">s2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:11_1"><span class="lab"><font color="Green" title="E21">A10</font></span></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></span>;<br/><a NAME="E31:11_1"/>
<font color="Maroon" title="c11">R</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</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>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="11_1_4"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c14">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c14">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">R</font> or <font color="Maroon" title="c14">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1</span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> )</span><br/>

<span class="kw">assume </span><a NAME="E1:11_1_4"/>
<font color="Maroon" title="c14">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">R</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c14">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1</span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span></span><br/>

<a NAME="E2:11_1_4"/><span class="kw">then </span>
 ex <font color="Olive" title="b1">s</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> st <br/>( <font color="Olive" title="b1">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">x</font> &amp; 1 <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Olive" title="b1">s</font> &amp; <font color="Olive" title="b1">s</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c10">s2</font> &amp; ( for <font color="Olive" title="b2">q</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="Olive" title="b2">q</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="Olive" title="b1">s</font> holds <br/><font color="Olive" title="b2">q</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">e</font> ) )
 
;<br/>
<span class="kw">hence </span><a NAME="E3:11_1_4"/>
<font color="Maroon" title="c14">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1</span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E30:11_1"><span class="lab"><font color="Green" title="E31">A21</font></span></a>, <a class="ref" href="xxreal_1.html#T1" title="XXREAL_1:th.1">XXREAL_1:1</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><a NAME="E32:11_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E32">A22</font></span>: 
1 <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c12">s0</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:11_1"><span class="lab"><font color="Green" title="E25">A14</font></span></a>, <a class="ref" href="borsuk_1.html#T83" title="BORSUK_1:th.83">BORSUK_1:83</a>, <a class="ref" href="borsuk_4.html#T51" title="BORSUK_4:th.51">BORSUK_4:51</a></span>;<br/><a NAME="E33:11_1"/><span class="kw">then </span>
<font color="Maroon" title="c12">s0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c8">f</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:11_1"><span class="lab"><font color="Green" title="E23">A12</font></span></a>, <a class="txt" href="#E30:11_1"><span class="lab"><font color="Green" title="E31">A21</font></span></a>, <a class="txt" href="#E23:11_1"><span class="lab"><font color="Green" title="E27">A17</font></span></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></span>;<br/><a NAME="E34:11_1"/><span class="kw">then </span>
<font color="Maroon" title="c8">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">s0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c8">f</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></span>;<br/><a NAME="E35:11_1"/><span class="kw">then </span>
<font color="Maroon" title="c8">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">s0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">P</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:11_1"><span class="lab"><font color="Green" title="E20">A9</font></span></a>, <a class="ref" href="pre_topc.html#D10" title="PRE_TOPC:def.10">PRE_TOPC:def 10</a></span>;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c14">p9</font> = <font color="Maroon" title="c8">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">s0</font> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> ;<br/><a NAME="E37:11_1"/><span class="lab"><font color="Green" title="E33">A23</font></span>: 
 <a href="jordan5c.html#R1" title="JORDAN5C:pred.1">LE</a> <font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c14">p9</font>,<font color="Maroon" title="c1">P</font>,<font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:11"><span class="lab"><font color="Green" title="E12">A1</font></span></a>, <a class="txt" href="#E4:11_1"><span class="lab"><font color="Green" title="E16">A5</font></span></a>, <a class="txt" href="#E5:11_1"><span class="lab"><font color="Green" title="E17">A6</font></span></a>, <a class="txt" href="#E6:11_1"><span class="lab"><font color="Green" title="E18">A7</font></span></a>, <a class="txt" href="#E12:11_1"><span class="lab"><font color="Green" title="E22">A11</font></span></a>, <a class="txt" href="#E30:11_1"><span class="lab"><font color="Green" title="E31">A21</font></span></a>, <a class="txt" href="#E15:11_1"><span class="lab"><font color="Green" title="E24">A13</font></span></a>, <a class="txt" href="#E32:11_1"><span class="lab"><font color="Green" title="E32">A22</font></span></a>, <a class="txt" href="#E23:11_1"><span class="lab"><font color="Green" title="E27">A17</font></span></a>, <a class="ref" href="jordan5c.html#T8" title="JORDAN5C:th.8">JORDAN5C:8</a></span>;<br/><a NAME="E38:11_1"/><span class="lab"><font color="Green" title="E34">A24</font></span>: 
 <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c8">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">P</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:11_1"><span class="lab"><font color="Green" title="E20">A9</font></span></a>, <a class="ref" href="pre_topc.html#D10" title="PRE_TOPC:def.10">PRE_TOPC:def 10</a></span>;<br/><a NAME="E39:11_1"/><span class="lab"><font color="Green" title="E35">A25</font></span>: 
for <font color="Olive" title="b1">p8</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  <a href="jordan5c.html#R1" title="JORDAN5C:pred.1">LE</a> <font color="Olive" title="b1">p8</font>,<font color="Maroon" title="c14">p9</font>,<font color="Maroon" title="c1">P</font>,<font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font> &amp;  <a href="jordan5c.html#R1" title="JORDAN5C:pred.1">LE</a> <font color="Maroon" title="c4">p</font>,<font color="Olive" title="b1">p8</font>,<font color="Maroon" title="c1">P</font>,<font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font> holds <br/><font color="Olive" title="b1">p8</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">e</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="11_1_5"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c15">p8</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  <a href="jordan5c.html#R1" title="JORDAN5C:pred.1">LE</a> <font color="Maroon" title="c15">p8</font>,<font color="Maroon" title="c14">p9</font>,<font color="Maroon" title="c1">P</font>,<font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font> &amp;  <a href="jordan5c.html#R1" title="JORDAN5C:pred.1">LE</a> <font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c15">p8</font>,<font color="Maroon" title="c1">P</font>,<font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font> implies <font color="Maroon" title="c15">p8</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">e</font> )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:11_1_5"/><span class="lab"><font color="Green" title="E36">A26</font></span>: 
 <a href="jordan5c.html#R1" title="JORDAN5C:pred.1">LE</a> <font color="Maroon" title="c15">p8</font>,<font color="Maroon" title="c14">p9</font>,<font color="Maroon" title="c1">P</font>,<font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font>
 <span class="kw">and </span><br/><a NAME="E2:11_1_5"/><span class="lab"><font color="Green" title="E37">A27</font></span>: 
 <a href="jordan5c.html#R1" title="JORDAN5C:pred.1">LE</a> <font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c15">p8</font>,<font color="Maroon" title="c1">P</font>,<font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c15">p8</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">e</font></span><br/>

<a NAME="E3:11_1_5"/><span class="lab"><font color="Green" title="E38">A28</font></span>: 
<font color="Maroon" title="c15">p8</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">P</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:11_1_5"><span class="lab"><font color="Green" title="E36">A26</font></span></a>, <a class="ref" href="jordan5c.html#D3" title="JORDAN5C:def.3">JORDAN5C:def 3</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c16">x8</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E5:11_1_5"/><span class="lab"><font color="Green" title="E39">A29</font></span>: 
<font color="Maroon" title="c16">x8</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c8">f</font>
 <span class="kw">and </span><br/><a NAME="E6:11_1_5"/><span class="lab"><font color="Green" title="E40">A30</font></span>: 
<font color="Maroon" title="c15">p8</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="c16">x8</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E38:11_1"><span class="lab"><font color="Green" title="E34">A24</font></span></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c17">s8</font> = <font color="Maroon" title="c16">x8</font> as   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:11_1"><span class="lab"><font color="Green" title="E23">A12</font></span></a>, <a class="txt" href="#E5:11_1_5"><span class="lab"><font color="Green" title="E39">A29</font></span></a>, <a class="ref" href="borsuk_1.html#T83" title="BORSUK_1:th.83">BORSUK_1:83</a></span>;<br/>
<a NAME="E8:11_1_5"/><span class="lab"><font color="Green" title="E41">A31</font></span>: 
<font color="Maroon" title="c17">s8</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:11_1_5"><span class="lab"><font color="Green" title="E39">A29</font></span></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></span>;<br/>
<a NAME="E9:11_1_5"/>
 <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="c17">s8</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:11_1_5"><span class="lab"><font color="Green" title="E39">A29</font></span></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></span>;<br/>
<a NAME="E10:11_1_5"/><span class="kw">then </span><span class="lab"><font color="Green" title="E42">A32</font></span>: 
<font color="Maroon" title="c17">s8</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c10">s2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:11_1"><span class="lab"><font color="Green" title="E16">A5</font></span></a>, <a class="txt" href="#E5:11_1"><span class="lab"><font color="Green" title="E17">A6</font></span></a>, <a class="txt" href="#E6:11_1"><span class="lab"><font color="Green" title="E18">A7</font></span></a>, <a class="txt" href="#E12:11_1"><span class="lab"><font color="Green" title="E22">A11</font></span></a>, <a class="txt" href="#E15:11_1"><span class="lab"><font color="Green" title="E24">A13</font></span></a>, <a class="txt" href="#E2:11_1_5"><span class="lab"><font color="Green" title="E37">A27</font></span></a>, <a class="txt" href="#E6:11_1_5"><span class="lab"><font color="Green" title="E40">A30</font></span></a>, <a class="txt" href="#E8:11_1_5"><span class="lab"><font color="Green" title="E41">A31</font></span></a>, <a class="ref" href="jordan5c.html#D3" title="JORDAN5C:def.3">JORDAN5C:def 3</a></span>;<br/>
<a NAME="E11:11_1_5"/><span class="lab"><font color="Green" title="E43">A33</font></span>: 
<font color="Maroon" title="c12">s0</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c17">s8</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:11_1"><span class="lab"><font color="Green" title="E16">A5</font></span></a>, <a class="txt" href="#E5:11_1"><span class="lab"><font color="Green" title="E17">A6</font></span></a>, <a class="txt" href="#E6:11_1"><span class="lab"><font color="Green" title="E18">A7</font></span></a>, <a class="txt" href="#E30:11_1"><span class="lab"><font color="Green" title="E31">A21</font></span></a>, <a class="txt" href="#E32:11_1"><span class="lab"><font color="Green" title="E32">A22</font></span></a>, <a class="txt" href="#E23:11_1"><span class="lab"><font color="Green" title="E27">A17</font></span></a>, <a class="txt" href="#E1:11_1_5"><span class="lab"><font color="Green" title="E36">A26</font></span></a>, <a class="txt" href="#E6:11_1_5"><span class="lab"><font color="Green" title="E40">A30</font></span></a>, <a class="txt" href="#E8:11_1_5"><span class="lab"><font color="Green" title="E41">A31</font></span></a>, <a class="ref" href="jordan5c.html#D3" title="JORDAN5C:def.3">JORDAN5C:def 3</a></span>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="11_1_5_1"><span class="kw">now </span></a><div class="add"><span class="kw">reconsider </span><font color="Maroon" title="c18">s8n</font> = <font color="Maroon" title="c17">s8</font> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <a href="metric_1.html#K8" title="METRIC_1:func.8">RealSpace</a>  <span class="kw">by</span> <span class="lab"><a class="ref" href="metric_1.html#D14" title="METRIC_1:def.14">METRIC_1:def 14</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c19">s8m</font> = <font color="Maroon" title="c17">s8</font> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K2" title="TOPMETR:func.2">Closed-Interval-MSpace</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1</span>)</span> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:11_1_5"><span class="lab"><font color="Green" title="E39">A29</font></span></a>, <a class="ref" href="borsuk_1.html#T83" title="BORSUK_1:th.83">BORSUK_1:83</a>, <a class="ref" href="topmetr.html#T14" title="TOPMETR:th.14">TOPMETR:14</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c20">ee</font> = <span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c15">p8</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c5">e</font></span>)</span></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2 as   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> ;<br/><span class="kw">reconsider </span><font color="Maroon" title="c21">w</font> = <font color="Maroon" title="c15">p8</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <a href="metric_1.html#K8" title="METRIC_1:func.8">RealSpace</a>  <span class="kw">by</span> <span class="lab"><a class="ref" href="metric_1.html#D14" title="METRIC_1:def.14">METRIC_1:def 14</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c22">B</font> =  <a href="metric_1.html#K9" title="METRIC_1:func.9">Ball</a> <font color="Maroon" title="c21">w</font>,<font color="Maroon" title="c20">ee</font> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <a href="topmetr.html#K3" title="TOPMETR:func.3">R^1</a>  <span class="kw">by</span> <span class="lab"><a class="ref" href="metric_1.html#D14" title="METRIC_1:def.14">METRIC_1:def 14</a>, <a class="ref" href="topmetr.html#T24" title="TOPMETR:th.24">TOPMETR:24</a></span>;<br/><span class="lab"><font color="Green" title="E44">A34</font></span>: <font color="Maroon" title="c22">B</font> = 
<span class="p1">{<span class="default"> <font color="Olive" title="b1">s7</font> where <font color="Olive" title="b1">s7</font> is   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> : ( <span class="p1">(<span class="default"><font color="Maroon" title="c15">p8</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c20">ee</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">s7</font> &amp; <font color="Olive" title="b1">s7</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">p8</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c20">ee</font> ) </span>}</span> 
<span class="kw">by</span> <span class="lab"><a class="ref" href="jordan2b.html#T21" title="JORDAN2B:th.21">JORDAN2B:21</a></span>
<br/>.= 
<span class="p1"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c15">p8</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c20">ee</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c15">p8</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c20">ee</font></span>)</span></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="rcomp_1.html#D2" title="RCOMP_1:def.2">RCOMP_1:def 2</a></span>
;<br/><span class="kw">assume </span><a NAME="E7:11_1_5_1"/><span class="lab"><font color="Green" title="E45">A35</font></span>: 
<font color="Maroon" title="c15">p8</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">e</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><a NAME="E8:11_1_5_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c15">p8</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c5">e</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ;<br/><a NAME="E9:11_1_5_1"/><span class="kw">then </span>
 <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c15">p8</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c5">e</font></span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="complex1.html#T133" title="COMPLEX1:th.133">COMPLEX1:133</a></span>;<br/><a NAME="E10:11_1_5_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E46">A36</font></span>: 
<font color="Maroon" title="c21">w</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="metric_1.html#K9" title="METRIC_1:func.9">Ball</a> <font color="Maroon" title="c21">w</font>,<font color="Maroon" title="c20">ee</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="goboard6.html#T4" title="GOBOARD6:th.4">GOBOARD6:4</a>, <a class="ref" href="xreal_1.html#T141" title="XREAL_1:th.141">XREAL_1:141</a></span>;<br/><a NAME="E11:11_1_5_1"/><span class="lab"><font color="Green" title="E47">A37</font></span>: 
( <font color="Maroon" title="c13">h</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</a> <font color="Maroon" title="c22">B</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  &amp;  <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pcomps_1.html#K3" title="PCOMPS_1:func.3">TopSpaceMetr</a> <span class="p1">(<span class="default"><a href="topmetr.html#K2" title="TOPMETR:func.2">Closed-Interval-MSpace</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1</span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:11_1"><span class="lab"><font color="Green" title="E19">A8</font></span></a>, <a class="txt" href="#E25:11_1"><span class="lab"><font color="Green" title="E28">A18</font></span></a>, <a class="ref" href="topmetr.html#T27" title="TOPMETR:th.27">TOPMETR:27</a>, <a class="ref" href="topmetr.html#D7" title="TOPMETR:def.7">TOPMETR:def 7</a>, <a class="ref" href="topmetr.html#D8" title="TOPMETR:def.8">TOPMETR:def 8</a>, <a class="ref" href="uniform1.html#T3" title="UNIFORM1:th.3">UNIFORM1:3</a></span>;<br/><font color="Maroon" title="c13">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">s8</font> = 
<font color="Maroon" title="c7">pro1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">s8</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E27:11_1"><span class="lab"><font color="Green" title="E29">A19</font></span></a>, <a class="txt" href="#E5:11_1_5"><span class="lab"><font color="Green" title="E39">A29</font></span></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#T22" title="FUNCT_1:th.22">FUNCT_1:22</a></span>
<br/>.= 
<a href="pscomp_1.html#K4" title="PSCOMP_1:func.4">proj1</a>  <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">p8</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:11_1_5"><span class="lab"><font color="Green" title="E38">A28</font></span></a>, <a class="txt" href="#E6:11_1_5"><span class="lab"><font color="Green" title="E40">A30</font></span></a>, <a class="ref" href="funct_1.html#T72" title="FUNCT_1:th.72">FUNCT_1:72</a></span>
<br/>.= 
<font color="Maroon" title="c15">p8</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> 
<span class="kw">by</span> <span class="lab"><a class="ref" href="pscomp_1.html#D28" title="PSCOMP_1:def.28">PSCOMP_1:def 28</a></span>
;<br/><a NAME="E13:11_1_5_1"/><span class="kw">then </span>
<font color="Maroon" title="c17">s8</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c13">h</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</a> <font color="Maroon" title="c22">B</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E27:11_1"><span class="lab"><font color="Green" title="E29">A19</font></span></a>, <a class="txt" href="#E5:11_1_5"><span class="lab"><font color="Green" title="E39">A29</font></span></a>, <a class="txt" href="#E10:11_1_5_1"><span class="lab"><font color="Green" title="E46">A36</font></span></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#D13" title="FUNCT_1:def.13">FUNCT_1:def 13</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c23">r0</font> being   <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>   <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> <span class="kw"> such that </span><br/><a NAME="E15:11_1_5_1"/><span class="lab"><font color="Green" title="E48">A38</font></span>: 
<font color="Maroon" title="c23">r0</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">and </span><br/><a NAME="E16:11_1_5_1"/><span class="lab"><font color="Green" title="E49">A39</font></span>: 
 <a href="metric_1.html#K9" title="METRIC_1:func.9">Ball</a> <font color="Maroon" title="c19">s8m</font>,<font color="Maroon" title="c23">r0</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c13">h</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</a> <font color="Maroon" title="c22">B</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:11_1_5_1"><span class="lab"><font color="Green" title="E47">A37</font></span></a>, <a class="ref" href="topmetr.html#T22" title="TOPMETR:th.22">TOPMETR:22</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c24">r0</font> = <font color="Maroon" title="c23">r0</font> as   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_0.html#D1" title="XREAL_0:def.1">XREAL_0:def 1</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c25">r01</font> =  <a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p1">(<span class="default"><font color="Maroon" title="c17">s8</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c10">s2</font></span>)</span>,<font color="Maroon" title="c24">r0</font> as   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> ;<br/><a NAME="E19:11_1_5_1"/>
( the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K2" title="TOPMETR:func.2">Closed-Interval-MSpace</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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> &amp;  <a href="metric_1.html#K9" title="METRIC_1:func.9">Ball</a> <font color="Maroon" title="c18">s8n</font>,<font color="Maroon" title="c25">r01</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c17">s8</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c25">r01</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c17">s8</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c25">r01</font></span>)</span></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="frechet.html#T7" title="FRECHET:th.7">FRECHET:7</a>, <a class="ref" href="topmetr.html#T14" title="TOPMETR:th.14">TOPMETR:14</a></span>;<br/><a NAME="E20:11_1_5_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E50">A40</font></span>: 
 <a href="metric_1.html#K9" title="METRIC_1:func.9">Ball</a> <font color="Maroon" title="c19">s8m</font>,<font color="Maroon" title="c25">r01</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c17">s8</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c25">r01</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c17">s8</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c25">r01</font></span>)</span></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1</span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="topmetr.html#T13" title="TOPMETR:th.13">TOPMETR:13</a></span>;<br/><a NAME="E21:11_1_5_1"/>
<font color="Maroon" title="c17">s8</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c10">s2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:11"><span class="lab"><font color="Green" title="E15">A4</font></span></a>, <a class="txt" href="#E12:11_1"><span class="lab"><font color="Green" title="E22">A11</font></span></a>, <a class="txt" href="#E6:11_1_5"><span class="lab"><font color="Green" title="E40">A30</font></span></a>, <a class="txt" href="#E10:11_1_5"><span class="lab"><font color="Green" title="E42">A32</font></span></a>, <a class="txt" href="#E7:11_1_5_1"><span class="lab"><font color="Green" title="E45">A35</font></span></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></span>;<br/><a NAME="E22:11_1_5_1"/><span class="kw">then </span>
<font color="Maroon" title="c17">s8</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c10">s2</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T52" title="XREAL_1:th.52">XREAL_1:52</a></span>;<br/><a NAME="E23:11_1_5_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E51">A41</font></span>: 
<font color="Maroon" title="c25">r01</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:11_1_5_1"><span class="lab"><font color="Green" title="E48">A38</font></span></a>, <a class="ref" href="xxreal_0.html#T21" title="XXREAL_0:th.21">XXREAL_0:21</a></span>;<br/><a NAME="E24:11_1_5_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E52">A42</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c25">r01</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c25">r01</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c25">r01</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c25">r01</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></span>;<br/><a NAME="E25:11_1_5_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E53">A43</font></span>: 
<font color="Maroon" title="c17">s8</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c25">r01</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c17">s8</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c25">r01</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T12" title="XREAL_1:th.12">XREAL_1:12</a></span>;<br/><a NAME="E26:11_1_5_1"/><span class="lab"><font color="Green" title="E54">A44</font></span>: 
<font color="Maroon" title="c25">r01</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E23:11_1_5_1"><span class="lab"><font color="Green" title="E51">A41</font></span></a>, <a class="ref" href="xreal_1.html#T141" title="XREAL_1:th.141">XREAL_1:141</a></span>;<br/><a NAME="E27:11_1_5_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E55">A45</font></span>: 
<font color="Maroon" title="c17">s8</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c25">r01</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c17">s8</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c25">r01</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T10" title="XREAL_1:th.10">XREAL_1:10</a></span>;<br/><a NAME="E28:11_1_5_1"/>
<font color="Maroon" title="c17">s8</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c25">r01</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c17">s8</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c25">r01</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E24:11_1_5_1"><span class="lab"><font color="Green" title="E52">A42</font></span></a>, <a class="ref" href="xreal_1.html#T10" title="XREAL_1:th.10">XREAL_1:10</a></span>;<br/><a NAME="E29:11_1_5_1"/><span class="kw">then </span>
<font color="Maroon" title="c17">s8</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c25">r01</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c17">s8</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c25">r01</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E27:11_1_5_1"><span class="lab"><font color="Green" title="E55">A45</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><a NAME="E30:11_1_5_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E56">A46</font></span>: 
<font color="Maroon" title="c17">s8</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c25">r01</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c17">s8</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c25">r01</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c17">s8</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c25">r01</font></span>)</span></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E25:11_1_5_1"><span class="lab"><font color="Green" title="E53">A43</font></span></a>, <a class="ref" href="xxreal_1.html#T4" title="XXREAL_1:th.4">XXREAL_1:4</a></span>;<br/><a NAME="E31:11_1_5_1"/><span class="lab"><font color="Green" title="E57">A47</font></span>: 
<font color="Maroon" title="c17">s8</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c25">r01</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c17">s8</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c25">r01</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E24:11_1_5_1"><span class="lab"><font color="Green" title="E52">A42</font></span></a>, <a class="ref" href="xreal_1.html#T12" title="XREAL_1:th.12">XREAL_1:12</a></span>;<br/><a NAME="E32:11_1_5_1"/>
 <a href="metric_1.html#K9" title="METRIC_1:func.9">Ball</a> <font color="Maroon" title="c19">s8m</font>,<font color="Maroon" title="c25">r01</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="metric_1.html#K9" title="METRIC_1:func.9">Ball</a> <font color="Maroon" title="c19">s8m</font>,<font color="Maroon" title="c24">r0</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="pcomps_1.html#T1" title="PCOMPS_1:th.1">PCOMPS_1:1</a>, <a class="ref" href="xxreal_0.html#T17" title="XXREAL_0:th.17">XXREAL_0:17</a></span>;<br/><a NAME="E33:11_1_5_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E58">A48</font></span>: 
<span class="p1"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c17">s8</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c25">r01</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c17">s8</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c25">r01</font></span>)</span></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</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> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c13">h</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</a> <font color="Maroon" title="c22">B</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:11_1_5_1"><span class="lab"><font color="Green" title="E49">A39</font></span></a>, <a class="txt" href="#E20:11_1_5_1"><span class="lab"><font color="Green" title="E50">A40</font></span></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c26">s70</font> = <font color="Maroon" title="c17">s8</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c25">r01</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span> as   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> ;<br/><a NAME="E35:11_1_5_1"/>
<font color="Maroon" title="c17">s8</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c10">s2</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c25">r01</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_0.html#T17" title="XXREAL_0:th.17">XXREAL_0:17</a></span>;<br/><a NAME="E36:11_1_5_1"/><span class="kw">then </span>
 <a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c17">s8</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c10">s2</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> <font color="Maroon" title="c25">r01</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T26" title="XREAL_1:th.26">XREAL_1:26</a></span>;<br/><a NAME="E37:11_1_5_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E59">A49</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c10">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c17">s8</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c17">s8</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> <font color="Maroon" title="c25">r01</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c17">s8</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T9" title="XREAL_1:th.9">XREAL_1:9</a></span>;<br/><a NAME="E38:11_1_5_1"/>
 <a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p1">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c25">r01</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E23:11_1_5_1"><span class="lab"><font color="Green" title="E51">A41</font></span></a>, <a class="ref" href="xreal_1.html#T141" title="XREAL_1:th.141">XREAL_1:141</a></span>;<br/><a NAME="E39:11_1_5_1"/><span class="kw">then </span>
 <a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c25">r01</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ;<br/><a NAME="E40:11_1_5_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E60">A50</font></span>: 
<font color="Maroon" title="c17">s8</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c17">s8</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c25">r01</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T10" title="XREAL_1:th.10">XREAL_1:10</a></span>;<br/><a NAME="E41:11_1_5_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E61">A51</font></span>: 
1 <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c26">s70</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:11_1_5"><span class="lab"><font color="Green" title="E41">A31</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><a NAME="E42:11_1_5_1"/>
1 <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c17">s8</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c25">r01</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:11_1_5"><span class="lab"><font color="Green" title="E41">A31</font></span></a>, <a class="txt" href="#E26:11_1_5_1"><span class="lab"><font color="Green" title="E54">A44</font></span></a>, <a class="ref" href="xreal_1.html#T17" title="XREAL_1:th.17">XREAL_1:17</a></span>;<br/><a NAME="E43:11_1_5_1"/><span class="kw">then </span>
<font color="Maroon" title="c17">s8</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c25">r01</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1</span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E30:11_1"><span class="lab"><font color="Green" title="E31">A21</font></span></a>, <a class="txt" href="#E37:11_1_5_1"><span class="lab"><font color="Green" title="E59">A49</font></span></a>, <a class="txt" href="#E31:11_1_5_1"><span class="lab"><font color="Green" title="E57">A47</font></span></a>, <a class="ref" href="xxreal_1.html#T1" title="XXREAL_1:th.1">XXREAL_1:1</a></span>;<br/><a NAME="E44:11_1_5_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E62">A52</font></span>: 
<font color="Maroon" title="c17">s8</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c25">r01</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c17">s8</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c25">r01</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c17">s8</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c25">r01</font></span>)</span></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1</span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E30:11_1_5_1"><span class="lab"><font color="Green" title="E56">A46</font></span></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>;<br/><a NAME="E45:11_1_5_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E63">A53</font></span>: 
<font color="Maroon" title="c13">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c17">s8</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c25">r01</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c22">B</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E33:11_1_5_1"><span class="lab"><font color="Green" title="E58">A48</font></span></a>, <a class="ref" href="funct_1.html#D13" title="FUNCT_1:def.13">FUNCT_1:def 13</a></span>;<br/><a NAME="E46:11_1_5_1"/><span class="lab"><font color="Green" title="E64">A54</font></span>: 
<font color="Maroon" title="c17">s8</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c25">r01</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c13">h</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E33:11_1_5_1"><span class="lab"><font color="Green" title="E58">A48</font></span></a>, <a class="txt" href="#E44:11_1_5_1"><span class="lab"><font color="Green" title="E62">A52</font></span></a>, <a class="ref" href="funct_1.html#D13" title="FUNCT_1:def.13">FUNCT_1:def 13</a></span>;<br/><a NAME="E47:11_1_5_1"/><span class="lab"><font color="Green" title="E65">A55</font></span>: 
for <font color="Olive" title="b1">p7</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="Olive" title="b1">p7</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="c26">s70</font> holds <br/><font color="Olive" title="b1">p7</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">e</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="11_1_5_1_3"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c27">p7</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c27">p7</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="c26">s70</font> implies <font color="Maroon" title="c27">p7</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">e</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:11_1_5_1_3"/><span class="lab"><font color="Green" title="E66">A56</font></span>: 
<font color="Maroon" title="c27">p7</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="c26">s70</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c27">p7</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">e</font></span><br/>

<a NAME="E2:11_1_5_1_3"/>
<font color="Maroon" title="c26">s70</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1</span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E30:11_1"><span class="lab"><font color="Green" title="E31">A21</font></span></a>, <a class="txt" href="#E37:11_1_5_1"><span class="lab"><font color="Green" title="E59">A49</font></span></a>, <a class="txt" href="#E25:11_1_5_1"><span class="lab"><font color="Green" title="E53">A43</font></span></a>, <a class="txt" href="#E41:11_1_5_1"><span class="lab"><font color="Green" title="E61">A51</font></span></a>, <a class="ref" href="xxreal_1.html#T1" title="XXREAL_1:th.1">XXREAL_1:1</a></span>;<br/>
<a NAME="E3:11_1_5_1_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E67">A57</font></span>: 
<font color="Maroon" title="c27">p7</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c8">f</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:11_1"><span class="lab"><font color="Green" title="E23">A12</font></span></a>, <a class="txt" href="#E1:11_1_5_1_3"><span class="lab"><font color="Green" title="E66">A56</font></span></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></span>;<br/>
<span class="lab"><font color="Green" title="E68">A58</font></span>:  <a href="relset_1.html#K2" title="RELSET_1:func.2">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">P</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:11_1"><span class="lab"><font color="Green" title="E16">A5</font></span></a>, <a class="ref" href="tops_2.html#D5" title="TOPS_2:def.5">TOPS_2:def 5</a></span>
<br/>.= 
<font color="Maroon" title="c1">P</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="pre_topc.html#D10" title="PRE_TOPC:def.10">PRE_TOPC:def 10</a></span>
;<br/>
<span class="lab"><font color="Green" title="E69">A59</font></span>: <font color="Maroon" title="c13">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c26">s70</font> = 
<font color="Maroon" title="c7">pro1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c26">s70</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E46:11_1_5_1"><span class="lab"><font color="Green" title="E64">A54</font></span></a>, <a class="ref" href="funct_1.html#T22" title="FUNCT_1:th.22">FUNCT_1:22</a></span>
<br/>.= 
<font color="Maroon" title="c6">pr1a</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c27">p7</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:11_1_5_1_3"><span class="lab"><font color="Green" title="E66">A56</font></span></a>, <a class="txt" href="#E3:11_1_5_1_3"><span class="lab"><font color="Green" title="E67">A57</font></span></a>, <a class="txt" href="#E4:11_1_5_1_3"><span class="lab"><font color="Green" title="E68">A58</font></span></a>, <a class="ref" href="funct_1.html#T72" title="FUNCT_1:th.72">FUNCT_1:72</a></span>
<br/>.= 
<font color="Maroon" title="c27">p7</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> 
<span class="kw">by</span> <span class="lab"><a class="ref" href="pscomp_1.html#D28" title="PSCOMP_1:def.28">PSCOMP_1:def 28</a></span>
;<br/>
<a NAME="E6:11_1_5_1_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E70">A60</font></span>: 
<font color="Maroon" title="c27">p7</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">p8</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c20">ee</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:11_1_5_1"><span class="lab"><font color="Green" title="E44">A34</font></span></a>, <a class="txt" href="#E45:11_1_5_1"><span class="lab"><font color="Green" title="E63">A53</font></span></a>, <a class="ref" href="xxreal_1.html#T4" title="XXREAL_1:th.4">XXREAL_1:4</a></span>;<br/>
<a NAME="E7:11_1_5_1_3"/><span class="lab"><font color="Green" title="E71">A61</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c15">p8</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c20">ee</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c27">p7</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:11_1_5_1"><span class="lab"><font color="Green" title="E44">A34</font></span></a>, <a class="txt" href="#E45:11_1_5_1"><span class="lab"><font color="Green" title="E63">A53</font></span></a>, <a class="txt" href="#E5:11_1_5_1_3"><span class="lab"><font color="Green" title="E69">A59</font></span></a>, <a class="ref" href="xxreal_1.html#T4" title="XXREAL_1:th.4">XXREAL_1:4</a></span>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="11_1_5_1_3_3"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:11_1_5_1_3_3"/><span class="lab"><font color="Green" title="E72">A62</font></span>: 
<font color="Maroon" title="c27">p7</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">e</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="11_1_5_1_3_3_1"><span class="kw">now </span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="11_1_5_1_3_3_1_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:11_1_5_1_3_3_1_1"/>
( <span class="p1">(<span class="default"><font color="Maroon" title="c15">p8</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c5">e</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  or <span class="p1">(<span class="default"><font color="Maroon" title="c15">p8</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c5">e</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="11_1_5_1_3_3_1_1_1"><span class="kw">case </span></a><a NAME="E1:11_1_5_1_3_3_1_1_1"/><span class="lab"><font color="Green" title="E73">A63</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c15">p8</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c5">e</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><a NAME="E2:11_1_5_1_3_3_1_1_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c15">p8</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c15">p8</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c5">e</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c5">e</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:11_1_5_1_3"><span class="lab"><font color="Green" title="E71">A61</font></span></a>, <a class="txt" href="#E1:11_1_5_1_3_3"><span class="lab"><font color="Green" title="E72">A62</font></span></a>, <a class="ref" href="absvalue.html#D1" title="ABSVALUE:def.1">ABSVALUE:def 1</a></span>;<br/><a NAME="E3:11_1_5_1_3_3_1_1_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c15">p8</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> </span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">e</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">e</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">e</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>
 ;<br/><a NAME="E4:11_1_5_1_3_3_1_1_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c15">p8</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> </span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c5">e</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T9" title="XREAL_1:th.9">XREAL_1:9</a></span>;<br/><a NAME="E5:11_1_5_1_3_3_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E74">A64</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c15">p8</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> </span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">e</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">e</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">e</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T16" title="XREAL_1:th.16">XREAL_1:16</a></span>;<br/><a NAME="E6:11_1_5_1_3_3_1_1_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c15">p8</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c5">e</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2 <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:11_1_5_1_3_3_1_1_1"><span class="lab"><font color="Green" title="E73">A63</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E7:11_1_5_1_3_3_1_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:11_1_5_1_3_3_1_1_1"><span class="lab"><font color="Green" title="E74">A64</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="11_1_5_1_3_3_1_1_2"><span class="kw">case </span></a><a NAME="E1:11_1_5_1_3_3_1_1_2"/><span class="lab"><font color="Green" title="E73">A65</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c15">p8</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c5">e</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><a NAME="E2:11_1_5_1_3_3_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c5">e</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">p8</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c15">p8</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c5">e</font></span>)</span></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:11_1_5_1_3"><span class="lab"><font color="Green" title="E70">A60</font></span></a>, <a class="txt" href="#E1:11_1_5_1_3_3"><span class="lab"><font color="Green" title="E72">A62</font></span></a>, <a class="ref" href="absvalue.html#D1" title="ABSVALUE:def.1">ABSVALUE:def 1</a></span>;<br/><a NAME="E3:11_1_5_1_3_3_1_1_2"/><span class="kw">then </span>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c15">p8</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> </span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">e</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">e</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">e</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>
 ;<br/><a NAME="E4:11_1_5_1_3_3_1_1_2"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c15">p8</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> </span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c5">e</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T9" title="XREAL_1:th.9">XREAL_1:9</a></span>;<br/><a NAME="E5:11_1_5_1_3_3_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E74">A66</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c15">p8</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> </span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">e</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">e</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">e</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T16" title="XREAL_1:th.16">XREAL_1:16</a></span>;<br/><a NAME="E6:11_1_5_1_3_3_1_1_2"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c15">p8</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c5">e</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:11_1_5_1_3_3_1_1_2"><span class="lab"><font color="Green" title="E73">A65</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E7:11_1_5_1_3_3_1_1_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:11_1_5_1_3_3_1_1_2"><span class="lab"><font color="Green" title="E74">A66</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E3:11_1_5_1_3_3"/>
contradiction
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E9:11_1_5_1_3"/>
<font color="Maroon" title="c27">p7</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">e</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><a NAME="E48:11_1_5_1"/>
<font color="Maroon" title="c26">s70</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c10">s2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E37:11_1_5_1"><span class="lab"><font color="Green" title="E59">A49</font></span></a>, <a class="txt" href="#E25:11_1_5_1"><span class="lab"><font color="Green" title="E53">A43</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c27">s7</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a><span class="kw"> such that </span><br/><a NAME="E50:11_1_5_1"/><span class="lab"><font color="Green" title="E66">A67</font></span>: 
<font color="Maroon" title="c17">s8</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c27">s7</font>
 <span class="kw">and </span><br/><a NAME="E51:11_1_5_1"/><span class="lab"><font color="Green" title="E67">A68</font></span>: 
( 1 <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c27">s7</font> &amp; <font color="Maroon" title="c27">s7</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c10">s2</font> &amp; ( for <font color="Olive" title="b1">p7</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="Olive" title="b1">p7</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="c27">s7</font> holds <br/><font color="Olive" title="b1">p7</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">e</font> ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E40:11_1_5_1"><span class="lab"><font color="Green" title="E60">A50</font></span></a>, <a class="txt" href="#E41:11_1_5_1"><span class="lab"><font color="Green" title="E61">A51</font></span></a>, <a class="txt" href="#E47:11_1_5_1"><span class="lab"><font color="Green" title="E65">A55</font></span></a></span>;<br/><a NAME="E52:11_1_5_1"/>
<font color="Maroon" title="c27">s7</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">R</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E51:11_1_5_1"><span class="lab"><font color="Green" title="E67">A68</font></span></a></span>;<br/><a NAME="E53:11_1_5_1"/><span class="kw">then </span>
<font color="Maroon" title="c27">s7</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c12">s0</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E29:11_1"><span class="lab"><font color="Green" title="E30">A20</font></span></a>, <a class="ref" href="seq_4.html#D5" title="SEQ_4:def.5">SEQ_4:def 5</a></span>;<br/><span class="kw">hence </span><a NAME="E54:11_1_5_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:11_1_5"><span class="lab"><font color="Green" title="E43">A33</font></span></a>, <a class="txt" href="#E50:11_1_5_1"><span class="lab"><font color="Green" title="E66">A67</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E13:11_1_5"/>
<font color="Maroon" title="c15">p8</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">e</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><span class="kw">assume </span><a NAME="E40:11_1"/>
not <font color="Maroon" title="c4">p</font> <a href="jordan20.html#R6" title="JORDAN20:pred.6">is_OSout</a> <font color="Maroon" title="c1">P</font>,<font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font>,<font color="Maroon" title="c5">e</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c4">p</font> <a href="jordan20.html#R3" title="JORDAN20:pred.3">is_Lout</a> <font color="Maroon" title="c1">P</font>,<font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font>,<font color="Maroon" title="c5">e</font> or <font color="Maroon" title="c4">p</font> <a href="jordan20.html#R4" title="JORDAN20:pred.4">is_Rout</a> <font color="Maroon" title="c1">P</font>,<font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font>,<font color="Maroon" title="c5">e</font> )</span><br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c15">p4</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span><span class="kw"> such that </span><br/><a NAME="E42:11_1"/><span class="lab"><font color="Green" title="E36">A69</font></span>: 
 <a href="jordan5c.html#R1" title="JORDAN5C:pred.1">LE</a> <font color="Maroon" title="c14">p9</font>,<font color="Maroon" title="c15">p4</font>,<font color="Maroon" title="c1">P</font>,<font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font>
 <span class="kw">and </span><br/><a NAME="E43:11_1"/><span class="lab"><font color="Green" title="E37">A70</font></span>: 
<font color="Maroon" title="c15">p4</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c14">p9</font>
 <span class="kw">and </span><br/><a NAME="E44:11_1"/><span class="lab"><font color="Green" title="E38">A71</font></span>: 
( for <font color="Olive" title="b1">p5</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  <a href="jordan5c.html#R1" title="JORDAN5C:pred.1">LE</a> <font color="Olive" title="b1">p5</font>,<font color="Maroon" title="c15">p4</font>,<font color="Maroon" title="c1">P</font>,<font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font> &amp;  <a href="jordan5c.html#R1" title="JORDAN5C:pred.1">LE</a> <font color="Maroon" title="c14">p9</font>,<font color="Olive" title="b1">p5</font>,<font color="Maroon" title="c1">P</font>,<font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font> holds <br/><font color="Olive" title="b1">p5</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">e</font> or for <font color="Olive" title="b1">p6</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  <a href="jordan5c.html#R1" title="JORDAN5C:pred.1">LE</a> <font color="Olive" title="b1">p6</font>,<font color="Maroon" title="c15">p4</font>,<font color="Maroon" title="c1">P</font>,<font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font> &amp;  <a href="jordan5c.html#R1" title="JORDAN5C:pred.1">LE</a> <font color="Maroon" title="c14">p9</font>,<font color="Olive" title="b1">p6</font>,<font color="Maroon" title="c1">P</font>,<font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font> holds <br/><font color="Olive" title="b1">p6</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c5">e</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:11"><span class="lab"><font color="Green" title="E12">A1</font></span></a>, <a class="txt" href="#E3:11"><span class="lab"><font color="Green" title="E14">A3</font></span></a>, <a class="txt" href="#E4:11"><span class="lab"><font color="Green" title="E15">A4</font></span></a>, <a class="txt" href="#E37:11_1"><span class="lab"><font color="Green" title="E33">A23</font></span></a>, <a class="txt" href="#E39:11_1"><span class="lab"><font color="Green" title="E35">A25</font></span></a>, <a class="ref" href="jordan20.html#D6" target="_self" title="JORDAN20:def.6">Def6</a></span>;<br/><a NAME="E45:11_1"/><span class="lab"><font color="Green" title="E39">A72</font></span>: 
<font color="Maroon" title="c14">p9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">P</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E42:11_1"><span class="lab"><font color="Green" title="E36">A69</font></span></a>, <a class="ref" href="jordan5c.html#D3" title="JORDAN5C:def.3">JORDAN5C:def 3</a></span>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="11_1_6"><span class="kw">now </span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="11_1_6_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:11_1_6_1"/>
( for <font color="Olive" title="b1">p5</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  <a href="jordan5c.html#R1" title="JORDAN5C:pred.1">LE</a> <font color="Olive" title="b1">p5</font>,<font color="Maroon" title="c15">p4</font>,<font color="Maroon" title="c1">P</font>,<font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font> &amp;  <a href="jordan5c.html#R1" title="JORDAN5C:pred.1">LE</a> <font color="Maroon" title="c14">p9</font>,<font color="Olive" title="b1">p5</font>,<font color="Maroon" title="c1">P</font>,<font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font> holds <br/><font color="Olive" title="b1">p5</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">e</font> or for <font color="Olive" title="b1">p6</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  <a href="jordan5c.html#R1" title="JORDAN5C:pred.1">LE</a> <font color="Olive" title="b1">p6</font>,<font color="Maroon" title="c15">p4</font>,<font color="Maroon" title="c1">P</font>,<font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font> &amp;  <a href="jordan5c.html#R1" title="JORDAN5C:pred.1">LE</a> <font color="Maroon" title="c14">p9</font>,<font color="Olive" title="b1">p6</font>,<font color="Maroon" title="c1">P</font>,<font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font> holds <br/><font color="Olive" title="b1">p6</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c5">e</font> )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E44:11_1"><span class="lab"><font color="Green" title="E38">A71</font></span></a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="11_1_6_1_1"><span class="kw">case </span></a><a NAME="E1:11_1_6_1_1"/><span class="lab"><font color="Green" title="E40">A73</font></span>: 
for <font color="Olive" title="b1">p5</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  <a href="jordan5c.html#R1" title="JORDAN5C:pred.1">LE</a> <font color="Olive" title="b1">p5</font>,<font color="Maroon" title="c15">p4</font>,<font color="Maroon" title="c1">P</font>,<font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font> &amp;  <a href="jordan5c.html#R1" title="JORDAN5C:pred.1">LE</a> <font color="Maroon" title="c14">p9</font>,<font color="Olive" title="b1">p5</font>,<font color="Maroon" title="c1">P</font>,<font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font> holds <br/><font color="Olive" title="b1">p5</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">e</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c4">p</font> <a href="jordan20.html#R3" title="JORDAN20:pred.3">is_Lout</a> <font color="Maroon" title="c1">P</font>,<font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font>,<font color="Maroon" title="c5">e</font> or <font color="Maroon" title="c4">p</font> <a href="jordan20.html#R4" title="JORDAN20:pred.4">is_Rout</a> <font color="Maroon" title="c1">P</font>,<font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font>,<font color="Maroon" title="c5">e</font> )</span><br/><div class="add"><div><span class="lab"><font color="Green" title="E41">A74</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="11_1_6_1_1_1"><span class="kw">now </span></a><div class="add"><a NAME="E1:11_1_6_1_1_1"/>
<font color="Maroon" title="c15">p4</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">P</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E42:11_1"><span class="lab"><font color="Green" title="E36">A69</font></span></a>, <a class="ref" href="jordan5c.html#D3" title="JORDAN5C:def.3">JORDAN5C:def 3</a></span>;<br/><a NAME="E2:11_1_6_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c15">p4</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c8">f</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:11_1"><span class="lab"><font color="Green" title="E20">A9</font></span></a>, <a class="ref" href="pre_topc.html#D10" title="PRE_TOPC:def.10">PRE_TOPC:def 10</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c16">xs4</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E4:11_1_6_1_1_1"/><span class="lab"><font color="Green" title="E42">A75</font></span>: 
<font color="Maroon" title="c16">xs4</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c8">f</font>
 <span class="kw">and </span><br/><a NAME="E5:11_1_6_1_1_1"/><span class="lab"><font color="Green" title="E43">A76</font></span>: 
<font color="Maroon" title="c15">p4</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="c16">xs4</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c17">s4</font> = <font color="Maroon" title="c16">xs4</font> as   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:11_1"><span class="lab"><font color="Green" title="E23">A12</font></span></a>, <a class="txt" href="#E4:11_1_6_1_1_1"><span class="lab"><font color="Green" title="E42">A75</font></span></a>, <a class="ref" href="borsuk_1.html#T83" title="BORSUK_1:th.83">BORSUK_1:83</a></span>;<br/><a NAME="E7:11_1_6_1_1_1"/><span class="lab"><font color="Green" title="E44">A77</font></span>: 
<font color="Maroon" title="c17">s4</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:11_1_6_1_1_1"><span class="lab"><font color="Green" title="E42">A75</font></span></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></span>;<br/><span class="kw">assume </span><a NAME="E8:11_1_6_1_1_1"/><span class="lab"><font color="Green" title="E45">A78</font></span>: 
for <font color="Olive" title="b1">p51</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>  holds <br/>(  not  <a href="jordan5c.html#R1" title="JORDAN5C:pred.1">LE</a> <font color="Olive" title="b1">p51</font>,<font color="Maroon" title="c15">p4</font>,<font color="Maroon" title="c1">P</font>,<font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font> or  not  <a href="jordan5c.html#R1" title="JORDAN5C:pred.1">LE</a> <font color="Maroon" title="c14">p9</font>,<font color="Olive" title="b1">p51</font>,<font color="Maroon" title="c1">P</font>,<font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font> or  not <font color="Olive" title="b1">p51</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c5">e</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><a NAME="E9:11_1_6_1_1_1"/><span class="lab"><font color="Green" title="E46">A79</font></span>: 
for <font color="Olive" title="b1">p51</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  <a href="jordan5c.html#R1" title="JORDAN5C:pred.1">LE</a> <font color="Olive" title="b1">p51</font>,<font color="Maroon" title="c15">p4</font>,<font color="Maroon" title="c1">P</font>,<font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font> &amp;  <a href="jordan5c.html#R1" title="JORDAN5C:pred.1">LE</a> <font color="Maroon" title="c14">p9</font>,<font color="Olive" title="b1">p51</font>,<font color="Maroon" title="c1">P</font>,<font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font> holds <br/><font color="Olive" title="b1">p51</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">e</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="11_1_6_1_1_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c18">p51</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  <a href="jordan5c.html#R1" title="JORDAN5C:pred.1">LE</a> <font color="Maroon" title="c18">p51</font>,<font color="Maroon" title="c15">p4</font>,<font color="Maroon" title="c1">P</font>,<font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font> &amp;  <a href="jordan5c.html#R1" title="JORDAN5C:pred.1">LE</a> <font color="Maroon" title="c14">p9</font>,<font color="Maroon" title="c18">p51</font>,<font color="Maroon" title="c1">P</font>,<font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font> implies <font color="Maroon" title="c18">p51</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">e</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:11_1_6_1_1_1_1"/>
(  <a href="jordan5c.html#R1" title="JORDAN5C:pred.1">LE</a> <font color="Maroon" title="c18">p51</font>,<font color="Maroon" title="c15">p4</font>,<font color="Maroon" title="c1">P</font>,<font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font> &amp;  <a href="jordan5c.html#R1" title="JORDAN5C:pred.1">LE</a> <font color="Maroon" title="c14">p9</font>,<font color="Maroon" title="c18">p51</font>,<font color="Maroon" title="c1">P</font>,<font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c18">p51</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">e</font></span><br/>

<a NAME="E2:11_1_6_1_1_1_1"/><span class="kw">then </span>
( <font color="Maroon" title="c18">p51</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c5">e</font> &amp; <font color="Maroon" title="c18">p51</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">e</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:11_1_6_1_1"><span class="lab"><font color="Green" title="E40">A73</font></span></a>, <a class="txt" href="#E8:11_1_6_1_1_1"><span class="lab"><font color="Green" title="E45">A78</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E3:11_1_6_1_1_1_1"/>
<font color="Maroon" title="c18">p51</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">e</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><div><span class="lab"><font color="Green" title="E47">A80</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="11_1_6_1_1_1_2"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:11_1_6_1_1_1_2"/>
<font color="Maroon" title="c17">s4</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c12">s0</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><a NAME="E2:11_1_6_1_1_1_2"/><span class="kw">then </span>
 <a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p1">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c17">s4</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c12">s0</font></span>)</span></span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T52" title="XREAL_1:th.52">XREAL_1:52</a></span>;<br/><a NAME="E3:11_1_6_1_1_1_2"/><span class="kw">then </span>
 <a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c17">s4</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c12">s0</font></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ;<br/><a NAME="E4:11_1_6_1_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E48">A81</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c12">s0</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c17">s4</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T143" title="XREAL_1:th.143">XREAL_1:143</a></span>;<br/><a NAME="E5:11_1_6_1_1_1_2"/><span class="kw">then </span>
 <a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c12">s0</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c17">s4</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T60" title="XREAL_1:th.60">XREAL_1:60</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c18">r</font> being   <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>   <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> <span class="kw"> such that </span><br/><a NAME="E7:11_1_6_1_1_1_2"/><span class="lab"><font color="Green" title="E49">A82</font></span>: 
<font color="Maroon" title="c18">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">R</font>
 <span class="kw">and </span><br/><a NAME="E8:11_1_6_1_1_1_2"/><span class="lab"><font color="Green" title="E50">A83</font></span>: 
<font color="Maroon" title="c18">r</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c12">s0</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c12">s0</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c17">s4</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E29:11_1"><span class="lab"><font color="Green" title="E30">A20</font></span></a>, <a class="ref" href="seq_4.html#D5" title="SEQ_4:def.5">SEQ_4:def 5</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c19">rss</font> = <font color="Maroon" title="c18">r</font> as   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_0.html#D1" title="XREAL_0:def.1">XREAL_0:def 1</a></span>;<br/><a NAME="E10:11_1_6_1_1_1_2"/><span class="lab"><font color="Green" title="E51">A84</font></span>: 
 ex <font color="Olive" title="b1">s7</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> st <br/>( <font color="Olive" title="b1">s7</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">r</font> &amp; 1 <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Olive" title="b1">s7</font> &amp; <font color="Olive" title="b1">s7</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c10">s2</font> &amp; ( for <font color="Olive" title="b2">q</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="Olive" title="b2">q</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="Olive" title="b1">s7</font> holds <br/><font color="Olive" title="b2">q</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">e</font> ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:11_1_6_1_1_1_2"><span class="lab"><font color="Green" title="E49">A82</font></span></a></span>;<br/><a NAME="E11:11_1_6_1_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c18">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1</span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E30:11_1"><span class="lab"><font color="Green" title="E31">A21</font></span></a>, <a class="ref" href="xxreal_1.html#T1" title="XXREAL_1:th.1">XXREAL_1:1</a></span>;<br/><a NAME="E12:11_1_6_1_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c8">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">rss</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c8">f</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:11_1"><span class="lab"><font color="Green" title="E23">A12</font></span></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></span>;<br/><a NAME="E13:11_1_6_1_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c8">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">rss</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">P</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:11_1"><span class="lab"><font color="Green" title="E20">A9</font></span></a>, <a class="ref" href="pre_topc.html#D10" title="PRE_TOPC:def.10">PRE_TOPC:def 10</a></span>;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c20">pss</font> = <font color="Maroon" title="c8">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">rss</font> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> ;<br/><a NAME="E15:11_1_6_1_1_1_2"/>
<font color="Maroon" title="c17">s4</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c17">s4</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c12">s0</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c17">s4</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:11_1_6_1_1_1_2"><span class="lab"><font color="Green" title="E48">A81</font></span></a>, <a class="ref" href="xreal_1.html#T10" title="XREAL_1:th.10">XREAL_1:10</a></span>;<br/><a NAME="E16:11_1_6_1_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E52">A85</font></span>: 
<font color="Maroon" title="c17">s4</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c18">r</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:11_1_6_1_1_1_2"><span class="lab"><font color="Green" title="E50">A83</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><a NAME="E17:11_1_6_1_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E53">A86</font></span>: 
1 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c18">r</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:11_1_6_1_1_1"><span class="lab"><font color="Green" title="E44">A77</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><a NAME="E18:11_1_6_1_1_1_2"/><span class="lab"><font color="Green" title="E54">A87</font></span>: 
<font color="Maroon" title="c18">r</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c12">s0</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E29:11_1"><span class="lab"><font color="Green" title="E30">A20</font></span></a>, <a class="txt" href="#E7:11_1_6_1_1_1_2"><span class="lab"><font color="Green" title="E49">A82</font></span></a>, <a class="ref" href="seq_4.html#D5" title="SEQ_4:def.5">SEQ_4:def 5</a></span>;<br/><a NAME="E19:11_1_6_1_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E55">A88</font></span>: 
 <a href="jordan5c.html#R1" title="JORDAN5C:pred.1">LE</a> <font color="Maroon" title="c14">p9</font>,<font color="Maroon" title="c20">pss</font>,<font color="Maroon" title="c1">P</font>,<font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:11"><span class="lab"><font color="Green" title="E12">A1</font></span></a>, <a class="txt" href="#E4:11_1"><span class="lab"><font color="Green" title="E16">A5</font></span></a>, <a class="txt" href="#E5:11_1"><span class="lab"><font color="Green" title="E17">A6</font></span></a>, <a class="txt" href="#E6:11_1"><span class="lab"><font color="Green" title="E18">A7</font></span></a>, <a class="txt" href="#E30:11_1"><span class="lab"><font color="Green" title="E31">A21</font></span></a>, <a class="txt" href="#E32:11_1"><span class="lab"><font color="Green" title="E32">A22</font></span></a>, <a class="txt" href="#E23:11_1"><span class="lab"><font color="Green" title="E27">A17</font></span></a>, <a class="txt" href="#E17:11_1_6_1_1_1_2"><span class="lab"><font color="Green" title="E53">A86</font></span></a>, <a class="ref" href="jordan5c.html#T8" title="JORDAN5C:th.8">JORDAN5C:8</a></span>;<br/><a NAME="E20:11_1_6_1_1_1_2"/>
 <a href="jordan5c.html#R1" title="JORDAN5C:pred.1">LE</a> <font color="Maroon" title="c20">pss</font>,<font color="Maroon" title="c15">p4</font>,<font color="Maroon" title="c1">P</font>,<font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:11"><span class="lab"><font color="Green" title="E12">A1</font></span></a>, <a class="txt" href="#E4:11_1"><span class="lab"><font color="Green" title="E16">A5</font></span></a>, <a class="txt" href="#E5:11_1"><span class="lab"><font color="Green" title="E17">A6</font></span></a>, <a class="txt" href="#E6:11_1"><span class="lab"><font color="Green" title="E18">A7</font></span></a>, <a class="txt" href="#E30:11_1"><span class="lab"><font color="Green" title="E31">A21</font></span></a>, <a class="txt" href="#E23:11_1"><span class="lab"><font color="Green" title="E27">A17</font></span></a>, <a class="txt" href="#E5:11_1_6_1_1_1"><span class="lab"><font color="Green" title="E43">A76</font></span></a>, <a class="txt" href="#E7:11_1_6_1_1_1"><span class="lab"><font color="Green" title="E44">A77</font></span></a>, <a class="txt" href="#E18:11_1_6_1_1_1_2"><span class="lab"><font color="Green" title="E54">A87</font></span></a>, <a class="txt" href="#E16:11_1_6_1_1_1_2"><span class="lab"><font color="Green" title="E52">A85</font></span></a>, <a class="txt" href="#E17:11_1_6_1_1_1_2"><span class="lab"><font color="Green" title="E53">A86</font></span></a>, <a class="ref" href="jordan5c.html#T8" title="JORDAN5C:th.8">JORDAN5C:8</a></span>;<br/><a NAME="E21:11_1_6_1_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c20">pss</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">e</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:11_1_6_1_1_1"><span class="lab"><font color="Green" title="E46">A79</font></span></a>, <a class="txt" href="#E19:11_1_6_1_1_1_2"><span class="lab"><font color="Green" title="E55">A88</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E22:11_1_6_1_1_1_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:11_1_6_1_1_1_2"><span class="lab"><font color="Green" title="E51">A84</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><a NAME="E11:11_1_6_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="c17">s4</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:11_1_6_1_1_1"><span class="lab"><font color="Green" title="E42">A75</font></span></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></span>;<br/><a NAME="E12:11_1_6_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c17">s4</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c12">s0</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:11_1"><span class="lab"><font color="Green" title="E16">A5</font></span></a>, <a class="txt" href="#E5:11_1"><span class="lab"><font color="Green" title="E17">A6</font></span></a>, <a class="txt" href="#E6:11_1"><span class="lab"><font color="Green" title="E18">A7</font></span></a>, <a class="txt" href="#E32:11_1"><span class="lab"><font color="Green" title="E32">A22</font></span></a>, <a class="txt" href="#E42:11_1"><span class="lab"><font color="Green" title="E36">A69</font></span></a>, <a class="txt" href="#E5:11_1_6_1_1_1"><span class="lab"><font color="Green" title="E43">A76</font></span></a>, <a class="txt" href="#E7:11_1_6_1_1_1"><span class="lab"><font color="Green" title="E44">A77</font></span></a>, <a class="ref" href="jordan5c.html#D3" title="JORDAN5C:def.3">JORDAN5C:def 3</a></span>;<br/><span class="kw">hence </span><a NAME="E13:11_1_6_1_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E43:11_1"><span class="lab"><font color="Green" title="E37">A70</font></span></a>, <a class="txt" href="#E5:11_1_6_1_1_1"><span class="lab"><font color="Green" title="E43">A76</font></span></a>, <a class="txt" href="#E10:11_1_6_1_1_1"><span class="lab"><font color="Green" title="E47">A80</font></span></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="11_1_6_1_1_2"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:11_1_6_1_1_2"/>
 ex <font color="Olive" title="b1">p51</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 <br/>(  <a href="jordan5c.html#R1" title="JORDAN5C:pred.1">LE</a> <font color="Olive" title="b1">p51</font>,<font color="Maroon" title="c15">p4</font>,<font color="Maroon" title="c1">P</font>,<font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font> &amp;  <a href="jordan5c.html#R1" title="JORDAN5C:pred.1">LE</a> <font color="Maroon" title="c14">p9</font>,<font color="Olive" title="b1">p51</font>,<font color="Maroon" title="c1">P</font>,<font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font> &amp; <font color="Olive" title="b1">p51</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c5">e</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c4">p</font> <a href="jordan20.html#R3" title="JORDAN20:pred.3">is_Lout</a> <font color="Maroon" title="c1">P</font>,<font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font>,<font color="Maroon" title="c5">e</font></span><br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c16">p51</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span><span class="kw"> such that </span><br/><a NAME="E3:11_1_6_1_1_2"/><span class="lab"><font color="Green" title="E42">A89</font></span>: 
 <a href="jordan5c.html#R1" title="JORDAN5C:pred.1">LE</a> <font color="Maroon" title="c16">p51</font>,<font color="Maroon" title="c15">p4</font>,<font color="Maroon" title="c1">P</font>,<font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font>
 <span class="kw">and </span><br/><a NAME="E4:11_1_6_1_1_2"/><span class="lab"><font color="Green" title="E43">A90</font></span>: 
 <a href="jordan5c.html#R1" title="JORDAN5C:pred.1">LE</a> <font color="Maroon" title="c14">p9</font>,<font color="Maroon" title="c16">p51</font>,<font color="Maroon" title="c1">P</font>,<font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font>
 <span class="kw">and </span><br/><a NAME="E5:11_1_6_1_1_2"/><span class="lab"><font color="Green" title="E44">A91</font></span>: 
<font color="Maroon" title="c16">p51</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c5">e</font>
 ;<br/><a NAME="E6:11_1_6_1_1_2"/><span class="lab"><font color="Green" title="E45">A92</font></span>: 
for <font color="Olive" title="b1">p5</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  <a href="jordan5c.html#R1" title="JORDAN5C:pred.1">LE</a> <font color="Olive" title="b1">p5</font>,<font color="Maroon" title="c16">p51</font>,<font color="Maroon" title="c1">P</font>,<font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font> &amp;  <a href="jordan5c.html#R1" title="JORDAN5C:pred.1">LE</a> <font color="Maroon" title="c4">p</font>,<font color="Olive" title="b1">p5</font>,<font color="Maroon" title="c1">P</font>,<font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font> holds <br/><font color="Olive" title="b1">p5</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">e</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="11_1_6_1_1_2_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c17">p5</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  <a href="jordan5c.html#R1" title="JORDAN5C:pred.1">LE</a> <font color="Maroon" title="c17">p5</font>,<font color="Maroon" title="c16">p51</font>,<font color="Maroon" title="c1">P</font>,<font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font> &amp;  <a href="jordan5c.html#R1" title="JORDAN5C:pred.1">LE</a> <font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c17">p5</font>,<font color="Maroon" title="c1">P</font>,<font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font> implies <font color="Maroon" title="c17">p5</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">e</font> )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:11_1_6_1_1_2_1"/><span class="lab"><font color="Green" title="E46">A93</font></span>: 
 <a href="jordan5c.html#R1" title="JORDAN5C:pred.1">LE</a> <font color="Maroon" title="c17">p5</font>,<font color="Maroon" title="c16">p51</font>,<font color="Maroon" title="c1">P</font>,<font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font>
 <span class="kw">and </span><br/><a NAME="E2:11_1_6_1_1_2_1"/><span class="lab"><font color="Green" title="E47">A94</font></span>: 
 <a href="jordan5c.html#R1" title="JORDAN5C:pred.1">LE</a> <font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c17">p5</font>,<font color="Maroon" title="c1">P</font>,<font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c17">p5</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">e</font></span><br/>

<a NAME="E3:11_1_6_1_1_2_1"/><span class="lab"><font color="Green" title="E48">A95</font></span>: 
 <a href="jordan5c.html#R1" title="JORDAN5C:pred.1">LE</a> <font color="Maroon" title="c17">p5</font>,<font color="Maroon" title="c15">p4</font>,<font color="Maroon" title="c1">P</font>,<font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:11_1_6_1_1_2"><span class="lab"><font color="Green" title="E42">A89</font></span></a>, <a class="txt" href="#E1:11_1_6_1_1_2_1"><span class="lab"><font color="Green" title="E46">A93</font></span></a>, <a class="ref" href="jordan5c.html#T13" title="JORDAN5C:th.13">JORDAN5C:13</a></span>;<br/>
<a NAME="E4:11_1_6_1_1_2_1"/><span class="lab"><font color="Green" title="E49">A96</font></span>: 
<font color="Maroon" title="c17">p5</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">P</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:11_1_6_1_1_2_1"><span class="lab"><font color="Green" title="E46">A93</font></span></a>, <a class="ref" href="jordan5c.html#D3" title="JORDAN5C:def.3">JORDAN5C:def 3</a></span>;<br/>
<a NAME="E5:11_1_6_1_1_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E50">A97</font></span>: 
( <font color="Maroon" title="c17">p5</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">p9</font> implies  <a href="jordan5c.html#R1" title="JORDAN5C:pred.1">LE</a> <font color="Maroon" title="c14">p9</font>,<font color="Maroon" title="c17">p5</font>,<font color="Maroon" title="c1">P</font>,<font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="jordan5c.html#T9" title="JORDAN5C:th.9">JORDAN5C:9</a></span>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="11_1_6_1_1_2_1_1"><span class="kw">now </span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="11_1_6_1_1_2_1_1_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:11_1_6_1_1_2_1_1_1"/>
(  <a href="jordan5c.html#R1" title="JORDAN5C:pred.1">LE</a> <font color="Maroon" title="c17">p5</font>,<font color="Maroon" title="c14">p9</font>,<font color="Maroon" title="c1">P</font>,<font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font> or  <a href="jordan5c.html#R1" title="JORDAN5C:pred.1">LE</a> <font color="Maroon" title="c14">p9</font>,<font color="Maroon" title="c17">p5</font>,<font color="Maroon" title="c1">P</font>,<font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font> )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:11"><span class="lab"><font color="Green" title="E12">A1</font></span></a>, <a class="txt" href="#E45:11_1"><span class="lab"><font color="Green" title="E39">A72</font></span></a>, <a class="txt" href="#E4:11_1_6_1_1_2_1"><span class="lab"><font color="Green" title="E49">A96</font></span></a>, <a class="txt" href="#E5:11_1_6_1_1_2_1"><span class="lab"><font color="Green" title="E50">A97</font></span></a>, <a class="ref" href="jordan5c.html#T14" title="JORDAN5C:th.14">JORDAN5C:14</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="11_1_6_1_1_2_1_1_1_1"><span class="kw">case </span></a><a NAME="E1:11_1_6_1_1_2_1_1_1_1"/>
 <a href="jordan5c.html#R1" title="JORDAN5C:pred.1">LE</a> <font color="Maroon" title="c17">p5</font>,<font color="Maroon" title="c14">p9</font>,<font color="Maroon" title="c1">P</font>,<font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c17">p5</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">e</font></span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:11_1_6_1_1_2_1_1_1_1"/>
<font color="Maroon" title="c17">p5</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">e</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E39:11_1"><span class="lab"><font color="Green" title="E35">A25</font></span></a>, <a class="txt" href="#E2:11_1_6_1_1_2_1"><span class="lab"><font color="Green" title="E47">A94</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="11_1_6_1_1_2_1_1_1_2"><span class="kw">case </span></a><a NAME="E1:11_1_6_1_1_2_1_1_1_2"/>
 <a href="jordan5c.html#R1" title="JORDAN5C:pred.1">LE</a> <font color="Maroon" title="c14">p9</font>,<font color="Maroon" title="c17">p5</font>,<font color="Maroon" title="c1">P</font>,<font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c17">p5</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">e</font></span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:11_1_6_1_1_2_1_1_1_2"/>
<font color="Maroon" title="c17">p5</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">e</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:11_1_6_1_1"><span class="lab"><font color="Green" title="E40">A73</font></span></a>, <a class="txt" href="#E3:11_1_6_1_1_2_1"><span class="lab"><font color="Green" title="E48">A95</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E7:11_1_6_1_1_2_1"/>
<font color="Maroon" title="c17">p5</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">e</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><a NAME="E7:11_1_6_1_1_2"/>
 <a href="jordan5c.html#R1" title="JORDAN5C:pred.1">LE</a> <font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c16">p51</font>,<font color="Maroon" title="c1">P</font>,<font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E37:11_1"><span class="lab"><font color="Green" title="E33">A23</font></span></a>, <a class="txt" href="#E4:11_1_6_1_1_2"><span class="lab"><font color="Green" title="E43">A90</font></span></a>, <a class="ref" href="jordan5c.html#T13" title="JORDAN5C:th.13">JORDAN5C:13</a></span>;<br/><span class="kw">hence </span><a NAME="E8:11_1_6_1_1_2"/>
<font color="Maroon" title="c4">p</font> <a href="jordan20.html#R3" title="JORDAN20:pred.3">is_Lout</a> <font color="Maroon" title="c1">P</font>,<font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font>,<font color="Maroon" title="c5">e</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:11"><span class="lab"><font color="Green" title="E12">A1</font></span></a>, <a class="txt" href="#E3:11"><span class="lab"><font color="Green" title="E14">A3</font></span></a>, <a class="txt" href="#E4:11"><span class="lab"><font color="Green" title="E15">A4</font></span></a>, <a class="txt" href="#E5:11_1_6_1_1_2"><span class="lab"><font color="Green" title="E44">A91</font></span></a>, <a class="txt" href="#E6:11_1_6_1_1_2"><span class="lab"><font color="Green" title="E45">A92</font></span></a>, <a class="ref" href="jordan20.html#D3" target="_self" title="JORDAN20:def.3">Def3</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E4:11_1_6_1_1"/>
( <font color="Maroon" title="c4">p</font> <a href="jordan20.html#R3" title="JORDAN20:pred.3">is_Lout</a> <font color="Maroon" title="c1">P</font>,<font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font>,<font color="Maroon" title="c5">e</font> or <font color="Maroon" title="c4">p</font> <a href="jordan20.html#R4" title="JORDAN20:pred.4">is_Rout</a> <font color="Maroon" title="c1">P</font>,<font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font>,<font color="Maroon" title="c5">e</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:11_1_6_1_1"><span class="lab"><font color="Green" title="E41">A74</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="11_1_6_1_2"><span class="kw">case </span></a><a NAME="E1:11_1_6_1_2"/><span class="lab"><font color="Green" title="E40">A98</font></span>: 
for <font color="Olive" title="b1">p6</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  <a href="jordan5c.html#R1" title="JORDAN5C:pred.1">LE</a> <font color="Olive" title="b1">p6</font>,<font color="Maroon" title="c15">p4</font>,<font color="Maroon" title="c1">P</font>,<font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font> &amp;  <a href="jordan5c.html#R1" title="JORDAN5C:pred.1">LE</a> <font color="Maroon" title="c14">p9</font>,<font color="Olive" title="b1">p6</font>,<font color="Maroon" title="c1">P</font>,<font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font> holds <br/><font color="Olive" title="b1">p6</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c5">e</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c4">p</font> <a href="jordan20.html#R3" title="JORDAN20:pred.3">is_Lout</a> <font color="Maroon" title="c1">P</font>,<font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font>,<font color="Maroon" title="c5">e</font> or <font color="Maroon" title="c4">p</font> <a href="jordan20.html#R4" title="JORDAN20:pred.4">is_Rout</a> <font color="Maroon" title="c1">P</font>,<font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font>,<font color="Maroon" title="c5">e</font> )</span><br/><div class="add"><div><span class="lab"><font color="Green" title="E41">A99</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="11_1_6_1_2_1"><span class="kw">now </span></a><div class="add"><a NAME="E1:11_1_6_1_2_1"/>
<font color="Maroon" title="c15">p4</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">P</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E42:11_1"><span class="lab"><font color="Green" title="E36">A69</font></span></a>, <a class="ref" href="jordan5c.html#D3" title="JORDAN5C:def.3">JORDAN5C:def 3</a></span>;<br/><a NAME="E2:11_1_6_1_2_1"/><span class="kw">then </span>
<font color="Maroon" title="c15">p4</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c8">f</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:11_1"><span class="lab"><font color="Green" title="E20">A9</font></span></a>, <a class="ref" href="pre_topc.html#D10" title="PRE_TOPC:def.10">PRE_TOPC:def 10</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c16">xs4</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E4:11_1_6_1_2_1"/><span class="lab"><font color="Green" title="E42">A100</font></span>: 
<font color="Maroon" title="c16">xs4</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c8">f</font>
 <span class="kw">and </span><br/><a NAME="E5:11_1_6_1_2_1"/><span class="lab"><font color="Green" title="E43">A101</font></span>: 
<font color="Maroon" title="c15">p4</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="c16">xs4</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c17">s4</font> = <font color="Maroon" title="c16">xs4</font> as   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:11_1"><span class="lab"><font color="Green" title="E23">A12</font></span></a>, <a class="txt" href="#E4:11_1_6_1_2_1"><span class="lab"><font color="Green" title="E42">A100</font></span></a>, <a class="ref" href="borsuk_1.html#T83" title="BORSUK_1:th.83">BORSUK_1:83</a></span>;<br/><a NAME="E7:11_1_6_1_2_1"/><span class="lab"><font color="Green" title="E44">A102</font></span>: 
<font color="Maroon" title="c17">s4</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:11_1_6_1_2_1"><span class="lab"><font color="Green" title="E42">A100</font></span></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></span>;<br/><span class="kw">assume </span><a NAME="E8:11_1_6_1_2_1"/><span class="lab"><font color="Green" title="E45">A103</font></span>: 
for <font color="Olive" title="b1">p51</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>  holds <br/>(  not  <a href="jordan5c.html#R1" title="JORDAN5C:pred.1">LE</a> <font color="Olive" title="b1">p51</font>,<font color="Maroon" title="c15">p4</font>,<font color="Maroon" title="c1">P</font>,<font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font> or  not  <a href="jordan5c.html#R1" title="JORDAN5C:pred.1">LE</a> <font color="Maroon" title="c14">p9</font>,<font color="Olive" title="b1">p51</font>,<font color="Maroon" title="c1">P</font>,<font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font> or  not <font color="Olive" title="b1">p51</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c5">e</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><a NAME="E9:11_1_6_1_2_1"/><span class="lab"><font color="Green" title="E46">A104</font></span>: 
for <font color="Olive" title="b1">p51</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  <a href="jordan5c.html#R1" title="JORDAN5C:pred.1">LE</a> <font color="Olive" title="b1">p51</font>,<font color="Maroon" title="c15">p4</font>,<font color="Maroon" title="c1">P</font>,<font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font> &amp;  <a href="jordan5c.html#R1" title="JORDAN5C:pred.1">LE</a> <font color="Maroon" title="c14">p9</font>,<font color="Olive" title="b1">p51</font>,<font color="Maroon" title="c1">P</font>,<font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font> holds <br/><font color="Olive" title="b1">p51</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">e</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="11_1_6_1_2_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c18">p51</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  <a href="jordan5c.html#R1" title="JORDAN5C:pred.1">LE</a> <font color="Maroon" title="c18">p51</font>,<font color="Maroon" title="c15">p4</font>,<font color="Maroon" title="c1">P</font>,<font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font> &amp;  <a href="jordan5c.html#R1" title="JORDAN5C:pred.1">LE</a> <font color="Maroon" title="c14">p9</font>,<font color="Maroon" title="c18">p51</font>,<font color="Maroon" title="c1">P</font>,<font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font> implies <font color="Maroon" title="c18">p51</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">e</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:11_1_6_1_2_1_1"/>
(  <a href="jordan5c.html#R1" title="JORDAN5C:pred.1">LE</a> <font color="Maroon" title="c18">p51</font>,<font color="Maroon" title="c15">p4</font>,<font color="Maroon" title="c1">P</font>,<font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font> &amp;  <a href="jordan5c.html#R1" title="JORDAN5C:pred.1">LE</a> <font color="Maroon" title="c14">p9</font>,<font color="Maroon" title="c18">p51</font>,<font color="Maroon" title="c1">P</font>,<font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c18">p51</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">e</font></span><br/>

<a NAME="E2:11_1_6_1_2_1_1"/><span class="kw">then </span>
( <font color="Maroon" title="c18">p51</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">e</font> &amp; <font color="Maroon" title="c18">p51</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c5">e</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:11_1_6_1_2"><span class="lab"><font color="Green" title="E40">A98</font></span></a>, <a class="txt" href="#E8:11_1_6_1_2_1"><span class="lab"><font color="Green" title="E45">A103</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E3:11_1_6_1_2_1_1"/>
<font color="Maroon" title="c18">p51</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">e</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><div><span class="lab"><font color="Green" title="E47">A105</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="11_1_6_1_2_1_2"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:11_1_6_1_2_1_2"/>
<font color="Maroon" title="c17">s4</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c12">s0</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><a NAME="E2:11_1_6_1_2_1_2"/><span class="kw">then </span>
 <a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p1">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c17">s4</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c12">s0</font></span>)</span></span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T52" title="XREAL_1:th.52">XREAL_1:52</a></span>;<br/><a NAME="E3:11_1_6_1_2_1_2"/><span class="kw">then </span>
 <a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c17">s4</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c12">s0</font></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ;<br/><a NAME="E4:11_1_6_1_2_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E48">A106</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c12">s0</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c17">s4</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T143" title="XREAL_1:th.143">XREAL_1:143</a></span>;<br/><a NAME="E5:11_1_6_1_2_1_2"/><span class="kw">then </span>
 <a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c12">s0</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c17">s4</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T60" title="XREAL_1:th.60">XREAL_1:60</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c18">r</font> being   <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>   <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> <span class="kw"> such that </span><br/><a NAME="E7:11_1_6_1_2_1_2"/><span class="lab"><font color="Green" title="E49">A107</font></span>: 
<font color="Maroon" title="c18">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">R</font>
 <span class="kw">and </span><br/><a NAME="E8:11_1_6_1_2_1_2"/><span class="lab"><font color="Green" title="E50">A108</font></span>: 
<font color="Maroon" title="c18">r</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c12">s0</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c12">s0</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c17">s4</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E29:11_1"><span class="lab"><font color="Green" title="E30">A20</font></span></a>, <a class="ref" href="seq_4.html#D5" title="SEQ_4:def.5">SEQ_4:def 5</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c19">rss</font> = <font color="Maroon" title="c18">r</font> as   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_0.html#D1" title="XREAL_0:def.1">XREAL_0:def 1</a></span>;<br/><a NAME="E10:11_1_6_1_2_1_2"/><span class="lab"><font color="Green" title="E51">A109</font></span>: 
 ex <font color="Olive" title="b1">s7</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> st <br/>( <font color="Olive" title="b1">s7</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">r</font> &amp; 1 <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Olive" title="b1">s7</font> &amp; <font color="Olive" title="b1">s7</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c10">s2</font> &amp; ( for <font color="Olive" title="b2">q</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="Olive" title="b2">q</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="Olive" title="b1">s7</font> holds <br/><font color="Olive" title="b2">q</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">e</font> ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:11_1_6_1_2_1_2"><span class="lab"><font color="Green" title="E49">A107</font></span></a></span>;<br/><a NAME="E11:11_1_6_1_2_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c18">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1</span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E30:11_1"><span class="lab"><font color="Green" title="E31">A21</font></span></a>, <a class="ref" href="xxreal_1.html#T1" title="XXREAL_1:th.1">XXREAL_1:1</a></span>;<br/><a NAME="E12:11_1_6_1_2_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c8">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">rss</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c8">f</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:11_1"><span class="lab"><font color="Green" title="E23">A12</font></span></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></span>;<br/><a NAME="E13:11_1_6_1_2_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c8">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">rss</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">P</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:11_1"><span class="lab"><font color="Green" title="E20">A9</font></span></a>, <a class="ref" href="pre_topc.html#D10" title="PRE_TOPC:def.10">PRE_TOPC:def 10</a></span>;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c20">pss</font> = <font color="Maroon" title="c8">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">rss</font> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> ;<br/><a NAME="E15:11_1_6_1_2_1_2"/>
<font color="Maroon" title="c17">s4</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c17">s4</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c12">s0</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c17">s4</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:11_1_6_1_2_1_2"><span class="lab"><font color="Green" title="E48">A106</font></span></a>, <a class="ref" href="xreal_1.html#T10" title="XREAL_1:th.10">XREAL_1:10</a></span>;<br/><a NAME="E16:11_1_6_1_2_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E52">A110</font></span>: 
<font color="Maroon" title="c17">s4</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c18">r</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:11_1_6_1_2_1_2"><span class="lab"><font color="Green" title="E50">A108</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><a NAME="E17:11_1_6_1_2_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E53">A111</font></span>: 
1 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c18">r</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:11_1_6_1_2_1"><span class="lab"><font color="Green" title="E44">A102</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><a NAME="E18:11_1_6_1_2_1_2"/><span class="lab"><font color="Green" title="E54">A112</font></span>: 
<font color="Maroon" title="c18">r</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c12">s0</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E29:11_1"><span class="lab"><font color="Green" title="E30">A20</font></span></a>, <a class="txt" href="#E7:11_1_6_1_2_1_2"><span class="lab"><font color="Green" title="E49">A107</font></span></a>, <a class="ref" href="seq_4.html#D5" title="SEQ_4:def.5">SEQ_4:def 5</a></span>;<br/><a NAME="E19:11_1_6_1_2_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E55">A113</font></span>: 
 <a href="jordan5c.html#R1" title="JORDAN5C:pred.1">LE</a> <font color="Maroon" title="c14">p9</font>,<font color="Maroon" title="c20">pss</font>,<font color="Maroon" title="c1">P</font>,<font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:11"><span class="lab"><font color="Green" title="E12">A1</font></span></a>, <a class="txt" href="#E4:11_1"><span class="lab"><font color="Green" title="E16">A5</font></span></a>, <a class="txt" href="#E5:11_1"><span class="lab"><font color="Green" title="E17">A6</font></span></a>, <a class="txt" href="#E6:11_1"><span class="lab"><font color="Green" title="E18">A7</font></span></a>, <a class="txt" href="#E30:11_1"><span class="lab"><font color="Green" title="E31">A21</font></span></a>, <a class="txt" href="#E32:11_1"><span class="lab"><font color="Green" title="E32">A22</font></span></a>, <a class="txt" href="#E23:11_1"><span class="lab"><font color="Green" title="E27">A17</font></span></a>, <a class="txt" href="#E17:11_1_6_1_2_1_2"><span class="lab"><font color="Green" title="E53">A111</font></span></a>, <a class="ref" href="jordan5c.html#T8" title="JORDAN5C:th.8">JORDAN5C:8</a></span>;<br/><a NAME="E20:11_1_6_1_2_1_2"/>
 <a href="jordan5c.html#R1" title="JORDAN5C:pred.1">LE</a> <font color="Maroon" title="c20">pss</font>,<font color="Maroon" title="c15">p4</font>,<font color="Maroon" title="c1">P</font>,<font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:11"><span class="lab"><font color="Green" title="E12">A1</font></span></a>, <a class="txt" href="#E4:11_1"><span class="lab"><font color="Green" title="E16">A5</font></span></a>, <a class="txt" href="#E5:11_1"><span class="lab"><font color="Green" title="E17">A6</font></span></a>, <a class="txt" href="#E6:11_1"><span class="lab"><font color="Green" title="E18">A7</font></span></a>, <a class="txt" href="#E30:11_1"><span class="lab"><font color="Green" title="E31">A21</font></span></a>, <a class="txt" href="#E23:11_1"><span class="lab"><font color="Green" title="E27">A17</font></span></a>, <a class="txt" href="#E5:11_1_6_1_2_1"><span class="lab"><font color="Green" title="E43">A101</font></span></a>, <a class="txt" href="#E7:11_1_6_1_2_1"><span class="lab"><font color="Green" title="E44">A102</font></span></a>, <a class="txt" href="#E18:11_1_6_1_2_1_2"><span class="lab"><font color="Green" title="E54">A112</font></span></a>, <a class="txt" href="#E16:11_1_6_1_2_1_2"><span class="lab"><font color="Green" title="E52">A110</font></span></a>, <a class="txt" href="#E17:11_1_6_1_2_1_2"><span class="lab"><font color="Green" title="E53">A111</font></span></a>, <a class="ref" href="jordan5c.html#T8" title="JORDAN5C:th.8">JORDAN5C:8</a></span>;<br/><a NAME="E21:11_1_6_1_2_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c20">pss</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">e</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:11_1_6_1_2_1"><span class="lab"><font color="Green" title="E46">A104</font></span></a>, <a class="txt" href="#E19:11_1_6_1_2_1_2"><span class="lab"><font color="Green" title="E55">A113</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E22:11_1_6_1_2_1_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:11_1_6_1_2_1_2"><span class="lab"><font color="Green" title="E51">A109</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><a NAME="E11:11_1_6_1_2_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="c17">s4</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:11_1_6_1_2_1"><span class="lab"><font color="Green" title="E42">A100</font></span></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></span>;<br/><a NAME="E12:11_1_6_1_2_1"/><span class="kw">then </span>
<font color="Maroon" title="c17">s4</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c12">s0</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:11_1"><span class="lab"><font color="Green" title="E16">A5</font></span></a>, <a class="txt" href="#E5:11_1"><span class="lab"><font color="Green" title="E17">A6</font></span></a>, <a class="txt" href="#E6:11_1"><span class="lab"><font color="Green" title="E18">A7</font></span></a>, <a class="txt" href="#E32:11_1"><span class="lab"><font color="Green" title="E32">A22</font></span></a>, <a class="txt" href="#E42:11_1"><span class="lab"><font color="Green" title="E36">A69</font></span></a>, <a class="txt" href="#E5:11_1_6_1_2_1"><span class="lab"><font color="Green" title="E43">A101</font></span></a>, <a class="txt" href="#E7:11_1_6_1_2_1"><span class="lab"><font color="Green" title="E44">A102</font></span></a>, <a class="ref" href="jordan5c.html#D3" title="JORDAN5C:def.3">JORDAN5C:def 3</a></span>;<br/><span class="kw">hence </span><a NAME="E13:11_1_6_1_2_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E43:11_1"><span class="lab"><font color="Green" title="E37">A70</font></span></a>, <a class="txt" href="#E5:11_1_6_1_2_1"><span class="lab"><font color="Green" title="E43">A101</font></span></a>, <a class="txt" href="#E10:11_1_6_1_2_1"><span class="lab"><font color="Green" title="E47">A105</font></span></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="11_1_6_1_2_2"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:11_1_6_1_2_2"/>
 ex <font color="Olive" title="b1">p51</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 <br/>(  <a href="jordan5c.html#R1" title="JORDAN5C:pred.1">LE</a> <font color="Olive" title="b1">p51</font>,<font color="Maroon" title="c15">p4</font>,<font color="Maroon" title="c1">P</font>,<font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font> &amp;  <a href="jordan5c.html#R1" title="JORDAN5C:pred.1">LE</a> <font color="Maroon" title="c14">p9</font>,<font color="Olive" title="b1">p51</font>,<font color="Maroon" title="c1">P</font>,<font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font> &amp; <font color="Olive" title="b1">p51</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c5">e</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c4">p</font> <a href="jordan20.html#R4" title="JORDAN20:pred.4">is_Rout</a> <font color="Maroon" title="c1">P</font>,<font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font>,<font color="Maroon" title="c5">e</font></span><br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c16">p51</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span><span class="kw"> such that </span><br/><a NAME="E3:11_1_6_1_2_2"/><span class="lab"><font color="Green" title="E42">A114</font></span>: 
 <a href="jordan5c.html#R1" title="JORDAN5C:pred.1">LE</a> <font color="Maroon" title="c16">p51</font>,<font color="Maroon" title="c15">p4</font>,<font color="Maroon" title="c1">P</font>,<font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font>
 <span class="kw">and </span><br/><a NAME="E4:11_1_6_1_2_2"/><span class="lab"><font color="Green" title="E43">A115</font></span>: 
 <a href="jordan5c.html#R1" title="JORDAN5C:pred.1">LE</a> <font color="Maroon" title="c14">p9</font>,<font color="Maroon" title="c16">p51</font>,<font color="Maroon" title="c1">P</font>,<font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font>
 <span class="kw">and </span><br/><a NAME="E5:11_1_6_1_2_2"/><span class="lab"><font color="Green" title="E44">A116</font></span>: 
<font color="Maroon" title="c16">p51</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c5">e</font>
 ;<br/><a NAME="E6:11_1_6_1_2_2"/><span class="lab"><font color="Green" title="E45">A117</font></span>: 
for <font color="Olive" title="b1">p5</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  <a href="jordan5c.html#R1" title="JORDAN5C:pred.1">LE</a> <font color="Olive" title="b1">p5</font>,<font color="Maroon" title="c16">p51</font>,<font color="Maroon" title="c1">P</font>,<font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font> &amp;  <a href="jordan5c.html#R1" title="JORDAN5C:pred.1">LE</a> <font color="Maroon" title="c4">p</font>,<font color="Olive" title="b1">p5</font>,<font color="Maroon" title="c1">P</font>,<font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font> holds <br/><font color="Olive" title="b1">p5</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c5">e</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="11_1_6_1_2_2_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c17">p5</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  <a href="jordan5c.html#R1" title="JORDAN5C:pred.1">LE</a> <font color="Maroon" title="c17">p5</font>,<font color="Maroon" title="c16">p51</font>,<font color="Maroon" title="c1">P</font>,<font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font> &amp;  <a href="jordan5c.html#R1" title="JORDAN5C:pred.1">LE</a> <font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c17">p5</font>,<font color="Maroon" title="c1">P</font>,<font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font> implies <font color="Maroon" title="c17">p5</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c5">e</font> )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:11_1_6_1_2_2_1"/><span class="lab"><font color="Green" title="E46">A118</font></span>: 
 <a href="jordan5c.html#R1" title="JORDAN5C:pred.1">LE</a> <font color="Maroon" title="c17">p5</font>,<font color="Maroon" title="c16">p51</font>,<font color="Maroon" title="c1">P</font>,<font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font>
 <span class="kw">and </span><br/><a NAME="E2:11_1_6_1_2_2_1"/><span class="lab"><font color="Green" title="E47">A119</font></span>: 
 <a href="jordan5c.html#R1" title="JORDAN5C:pred.1">LE</a> <font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c17">p5</font>,<font color="Maroon" title="c1">P</font>,<font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c17">p5</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c5">e</font></span><br/>

<a NAME="E3:11_1_6_1_2_2_1"/><span class="lab"><font color="Green" title="E48">A120</font></span>: 
 <a href="jordan5c.html#R1" title="JORDAN5C:pred.1">LE</a> <font color="Maroon" title="c17">p5</font>,<font color="Maroon" title="c15">p4</font>,<font color="Maroon" title="c1">P</font>,<font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:11_1_6_1_2_2"><span class="lab"><font color="Green" title="E42">A114</font></span></a>, <a class="txt" href="#E1:11_1_6_1_2_2_1"><span class="lab"><font color="Green" title="E46">A118</font></span></a>, <a class="ref" href="jordan5c.html#T13" title="JORDAN5C:th.13">JORDAN5C:13</a></span>;<br/>
<a NAME="E4:11_1_6_1_2_2_1"/><span class="lab"><font color="Green" title="E49">A121</font></span>: 
<font color="Maroon" title="c17">p5</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">P</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:11_1_6_1_2_2_1"><span class="lab"><font color="Green" title="E46">A118</font></span></a>, <a class="ref" href="jordan5c.html#D3" title="JORDAN5C:def.3">JORDAN5C:def 3</a></span>;<br/>
<a NAME="E5:11_1_6_1_2_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E50">A122</font></span>: 
( <font color="Maroon" title="c17">p5</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">p9</font> implies  <a href="jordan5c.html#R1" title="JORDAN5C:pred.1">LE</a> <font color="Maroon" title="c14">p9</font>,<font color="Maroon" title="c17">p5</font>,<font color="Maroon" title="c1">P</font>,<font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="jordan5c.html#T9" title="JORDAN5C:th.9">JORDAN5C:9</a></span>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="11_1_6_1_2_2_1_1"><span class="kw">now </span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="11_1_6_1_2_2_1_1_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:11_1_6_1_2_2_1_1_1"/>
(  <a href="jordan5c.html#R1" title="JORDAN5C:pred.1">LE</a> <font color="Maroon" title="c14">p9</font>,<font color="Maroon" title="c17">p5</font>,<font color="Maroon" title="c1">P</font>,<font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font> or  <a href="jordan5c.html#R1" title="JORDAN5C:pred.1">LE</a> <font color="Maroon" title="c17">p5</font>,<font color="Maroon" title="c14">p9</font>,<font color="Maroon" title="c1">P</font>,<font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font> )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:11"><span class="lab"><font color="Green" title="E12">A1</font></span></a>, <a class="txt" href="#E45:11_1"><span class="lab"><font color="Green" title="E39">A72</font></span></a>, <a class="txt" href="#E4:11_1_6_1_2_2_1"><span class="lab"><font color="Green" title="E49">A121</font></span></a>, <a class="txt" href="#E5:11_1_6_1_2_2_1"><span class="lab"><font color="Green" title="E50">A122</font></span></a>, <a class="ref" href="jordan5c.html#T14" title="JORDAN5C:th.14">JORDAN5C:14</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="11_1_6_1_2_2_1_1_1_1"><span class="kw">case </span></a><a NAME="E1:11_1_6_1_2_2_1_1_1_1"/>
 <a href="jordan5c.html#R1" title="JORDAN5C:pred.1">LE</a> <font color="Maroon" title="c14">p9</font>,<font color="Maroon" title="c17">p5</font>,<font color="Maroon" title="c1">P</font>,<font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c17">p5</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c5">e</font></span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:11_1_6_1_2_2_1_1_1_1"/>
<font color="Maroon" title="c17">p5</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c5">e</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:11_1_6_1_2"><span class="lab"><font color="Green" title="E40">A98</font></span></a>, <a class="txt" href="#E3:11_1_6_1_2_2_1"><span class="lab"><font color="Green" title="E48">A120</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="11_1_6_1_2_2_1_1_1_2"><span class="kw">case </span></a><a NAME="E1:11_1_6_1_2_2_1_1_1_2"/>
 <a href="jordan5c.html#R1" title="JORDAN5C:pred.1">LE</a> <font color="Maroon" title="c17">p5</font>,<font color="Maroon" title="c14">p9</font>,<font color="Maroon" title="c1">P</font>,<font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c17">p5</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c5">e</font></span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:11_1_6_1_2_2_1_1_1_2"/>
<font color="Maroon" title="c17">p5</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c5">e</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E39:11_1"><span class="lab"><font color="Green" title="E35">A25</font></span></a>, <a class="txt" href="#E2:11_1_6_1_2_2_1"><span class="lab"><font color="Green" title="E47">A119</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E7:11_1_6_1_2_2_1"/>
<font color="Maroon" title="c17">p5</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c5">e</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><a NAME="E7:11_1_6_1_2_2"/>
 <a href="jordan5c.html#R1" title="JORDAN5C:pred.1">LE</a> <font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c16">p51</font>,<font color="Maroon" title="c1">P</font>,<font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E37:11_1"><span class="lab"><font color="Green" title="E33">A23</font></span></a>, <a class="txt" href="#E4:11_1_6_1_2_2"><span class="lab"><font color="Green" title="E43">A115</font></span></a>, <a class="ref" href="jordan5c.html#T13" title="JORDAN5C:th.13">JORDAN5C:13</a></span>;<br/><span class="kw">hence </span><a NAME="E8:11_1_6_1_2_2"/>
<font color="Maroon" title="c4">p</font> <a href="jordan20.html#R4" title="JORDAN20:pred.4">is_Rout</a> <font color="Maroon" title="c1">P</font>,<font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font>,<font color="Maroon" title="c5">e</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:11"><span class="lab"><font color="Green" title="E12">A1</font></span></a>, <a class="txt" href="#E3:11"><span class="lab"><font color="Green" title="E14">A3</font></span></a>, <a class="txt" href="#E4:11"><span class="lab"><font color="Green" title="E15">A4</font></span></a>, <a class="txt" href="#E5:11_1_6_1_2_2"><span class="lab"><font color="Green" title="E44">A116</font></span></a>, <a class="txt" href="#E6:11_1_6_1_2_2"><span class="lab"><font color="Green" title="E45">A117</font></span></a>, <a class="ref" href="jordan20.html#D4" target="_self" title="JORDAN20:def.4">Def4</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E4:11_1_6_1_2"/>
( <font color="Maroon" title="c4">p</font> <a href="jordan20.html#R3" title="JORDAN20:pred.3">is_Lout</a> <font color="Maroon" title="c1">P</font>,<font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font>,<font color="Maroon" title="c5">e</font> or <font color="Maroon" title="c4">p</font> <a href="jordan20.html#R4" title="JORDAN20:pred.4">is_Rout</a> <font color="Maroon" title="c1">P</font>,<font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font>,<font color="Maroon" title="c5">e</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:11_1_6_1_2"><span class="lab"><font color="Green" title="E41">A99</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E47:11_1"/>
( <font color="Maroon" title="c4">p</font> <a href="jordan20.html#R3" title="JORDAN20:pred.3">is_Lout</a> <font color="Maroon" title="c1">P</font>,<font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font>,<font color="Maroon" title="c5">e</font> or <font color="Maroon" title="c4">p</font> <a href="jordan20.html#R4" title="JORDAN20:pred.4">is_Rout</a> <font color="Maroon" title="c1">P</font>,<font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font>,<font color="Maroon" title="c5">e</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E6:11"/>
( <font color="Maroon" title="c4">p</font> <a href="jordan20.html#R3" title="JORDAN20:pred.3">is_Lout</a> <font color="Maroon" title="c1">P</font>,<font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font>,<font color="Maroon" title="c5">e</font> or <font color="Maroon" title="c4">p</font> <a href="jordan20.html#R4" title="JORDAN20:pred.4">is_Rout</a> <font color="Maroon" title="c1">P</font>,<font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font>,<font color="Maroon" title="c5">e</font> or <font color="Maroon" title="c4">p</font> <a href="jordan20.html#R6" title="JORDAN20:pred.6">is_OSout</a> <font color="Maroon" title="c1">P</font>,<font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font>,<font color="Maroon" title="c5">e</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
