<?xml version="1.0"?>
<div class="add">

<span class="kw">let </span><font color="Maroon" title="c1">n</font> be   <a href="ordinal1.html#V7" title="ORDINAL1:attr.7">natural</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="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">p</font>, <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> <span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span>  st <font color="Olive" title="b1">p</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span> holds <br/> <a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">n</font>, <a href="mfold_2.html#K3" title="MFOLD_2:func.3">TPlane</a> (<font color="Olive" title="b1">p</font>,<font color="Olive" title="b2">q</font>) <a href="mfold_2.html#R1" title="MFOLD_2:pred.1">are_homeomorphic</a> </span><br/>

<span class="kw">set </span><font color="Maroon" title="c2">T1</font> =  <a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span>;<br/>
<span class="kw">let </span><font color="Maroon" title="c3">p</font>, <font color="Maroon" title="c4">q</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> <span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c3">p</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span> implies  <a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">n</font>, <a href="mfold_2.html#K3" title="MFOLD_2:func.3">TPlane</a> (<font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c4">q</font>) <a href="mfold_2.html#R1" title="MFOLD_2:pred.1">are_homeomorphic</a>  )</span><br/>

<span class="kw">assume </span><a NAME="E1:38"/><span class="lab"><font color="Green" title="E32">A1</font></span>: 
<font color="Maroon" title="c3">p</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">n</font>, <a href="mfold_2.html#K3" title="MFOLD_2:func.3">TPlane</a> (<font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c4">q</font>) <a href="mfold_2.html#R1" title="MFOLD_2:pred.1">are_homeomorphic</a> </span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c5">p0</font> =  <a href="euclid_7.html#K6" title="EUCLID_7:func.6">Base_FinSeq</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span>) 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> <span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span> <span class="kw">by</span> <span class="lab"><a class="ref" href="euclid.html#T22" title="EUCLID:th.22">EUCLID:22</a></span>;<br/>
<a NAME="E3:38"/><span class="lab"><font color="Green" title="E33">A2</font></span>: 
<font color="Maroon" title="c5">p0</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="38_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:38_1"/><span class="lab"><font color="Green" title="E34">A3</font></span>: 
<font color="Maroon" title="c5">p0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<a NAME="E2:38_1"/>
<a href="card_1.html#NK1" title="CARD_1:NK.1">0</a> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">n</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T6" title="XREAL_1:th.6">XREAL_1:6</a></span>;<br/>
<a NAME="E3:38_1"/><span class="kw">then </span>
<span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><font color="Maroon" title="c5">p0</font></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="euclid_7.html#T28" title="EUCLID_7:th.28">EUCLID_7:28</a></span>;<br/>
<span class="kw">hence </span><a NAME="E4:38_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:38_1"><span class="lab"><font color="Green" title="E34">A3</font></span></a>, <a class="ref" href="euclid_2.html#T39" title="EUCLID_2:th.39">EUCLID_2:39</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E4:38"/><span class="lab"><font color="Green" title="E34">A4</font></span>: 
 <a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">n</font>, <a href="mfold_2.html#K3" title="MFOLD_2:func.3">TPlane</a> (<font color="Maroon" title="c5">p0</font>,<span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span></span>)</span>) <a href="mfold_2.html#R1" title="MFOLD_2:pred.1">are_homeomorphic</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="mfold_2.html#E34"><span class="lab"><font color="Green" title="E30">Lm3</font></span></a></span>;<br/>
<a NAME="E5:38"/>
 ex <font color="Olive" title="b1">R</font> being   <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> <span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span> st <br/>( <font color="Olive" title="b1">R</font> is  <a href="tops_2.html#V3" title="TOPS_2:attr.3">being_homeomorphism</a>  &amp; <font color="Olive" title="b1">R</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <span class="p1">(<span class="default"><a href="mfold_2.html#K2" title="MFOLD_2:func.2">Plane</a> (<font color="Maroon" title="c5">p0</font>,<span class="p2">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p3">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <span class="p4">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span></span>)</span>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="mfold_2.html#K2" title="MFOLD_2:func.2">Plane</a> (<font color="Maroon" title="c3">p</font>,<span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span></span>)</span>) )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:38"><span class="lab"><font color="Green" title="E32">A1</font></span></a>, <a class="txt" href="#E3:38"><span class="lab"><font color="Green" title="E33">A2</font></span></a>, <a class="ref" href="mfold_2.html#T27" target="_self" title="MFOLD_2:th.27">Th27</a></span>;<br/>
<a NAME="E6:38"/><span class="kw">then </span>
 <a href="mfold_2.html#K2" title="MFOLD_2:func.2">Plane</a> (<font color="Maroon" title="c5">p0</font>,<span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span></span>)</span>), <a href="mfold_2.html#K2" title="MFOLD_2:func.2">Plane</a> (<font color="Maroon" title="c3">p</font>,<span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span></span>)</span>) <a href="metrizts.html#R1" title="METRIZTS:pred.1">are_homeomorphic</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="metrizts.html#T3" title="METRIZTS:th.3">METRIZTS:3</a></span>;<br/>
<a NAME="E7:38"/><span class="kw">then </span>
 <a href="mfold_2.html#K3" title="MFOLD_2:func.3">TPlane</a> (<font color="Maroon" title="c5">p0</font>,<span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span></span>)</span>), <a href="mfold_2.html#K3" title="MFOLD_2:func.3">TPlane</a> (<font color="Maroon" title="c3">p</font>,<span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span></span>)</span>) <a href="mfold_2.html#R1" title="MFOLD_2:pred.1">are_homeomorphic</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="metrizts.html#D1" title="METRIZTS:def.1">METRIZTS:def 1</a></span>;<br/>
<a NAME="E8:38"/><span class="kw">then </span><span class="lab"><font color="Green" title="E35">A5</font></span>: 
 <a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">n</font>, <a href="mfold_2.html#K3" title="MFOLD_2:func.3">TPlane</a> (<font color="Maroon" title="c3">p</font>,<span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span></span>)</span>) <a href="mfold_2.html#R1" title="MFOLD_2:pred.1">are_homeomorphic</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:38"><span class="lab"><font color="Green" title="E34">A4</font></span></a>, <a class="ref" href="borsuk_3.html#T3" title="BORSUK_3:th.3">BORSUK_3:3</a></span>;<br/>
<span class="p1">(<span class="default"><a href="rltopsp1.html#K3" title="RLTOPSP1:func.3">transl</a> (<font color="Maroon" title="c4">q</font>,<span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span>)</span>)</span> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <span class="p1">(<span class="default"><a href="mfold_2.html#K2" title="MFOLD_2:func.2">Plane</a> (<font color="Maroon" title="c3">p</font>,<span class="p2">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p3">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <span class="p4">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span></span>)</span>)</span>)</span> = 
 <a href="mfold_2.html#K2" title="MFOLD_2:func.2">Plane</a> (<font color="Maroon" title="c3">p</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p3">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <span class="p4">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c4">q</font></span>)</span>)
<span class="kw">by</span> <span class="lab"><a class="ref" href="mfold_2.html#T25" target="_self" title="MFOLD_2:th.25">Th25</a></span>
<br/>.= 
 <a href="mfold_2.html#K2" title="MFOLD_2:func.2">Plane</a> (<font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c4">q</font>)
<span class="kw">by</span> <span class="lab"><a class="ref" href="euclid.html#T27" title="EUCLID:th.27">EUCLID:27</a></span>
;<br/>
<a NAME="E10:38"/><span class="kw">then </span>
 <a href="mfold_2.html#K2" title="MFOLD_2:func.2">Plane</a> (<font color="Maroon" title="c3">p</font>,<span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span></span>)</span>), <a href="mfold_2.html#K2" title="MFOLD_2:func.2">Plane</a> (<font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c4">q</font>) <a href="metrizts.html#R1" title="METRIZTS:pred.1">are_homeomorphic</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="metrizts.html#T3" title="METRIZTS:th.3">METRIZTS:3</a></span>;<br/>
<a NAME="E11:38"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <span class="p1">(<span class="default"><a href="mfold_2.html#K2" title="MFOLD_2:func.2">Plane</a> (<font color="Maroon" title="c3">p</font>,<span class="p2">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p3">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <span class="p4">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span></span>)</span>)</span>)</span>,<span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <span class="p1">(<span class="default"><a href="mfold_2.html#K2" title="MFOLD_2:func.2">Plane</a> (<font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c4">q</font>)</span>)</span> <a href="mfold_2.html#R1" title="MFOLD_2:pred.1">are_homeomorphic</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="metrizts.html#D1" title="METRIZTS:def.1">METRIZTS:def 1</a></span>;<br/>
<span class="kw">hence </span><a NAME="E12:38"/>
 <a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">n</font>, <a href="mfold_2.html#K3" title="MFOLD_2:func.3">TPlane</a> (<font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c4">q</font>) <a href="mfold_2.html#R1" title="MFOLD_2:pred.1">are_homeomorphic</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:38"><span class="lab"><font color="Green" title="E35">A5</font></span></a>, <a class="ref" href="borsuk_3.html#T3" title="BORSUK_3:th.3">BORSUK_3:3</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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