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<span class="kw">let </span><font color="Maroon" title="c1">n</font> be   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">P1</font>, <font color="Olive" title="b2">P2</font> being   <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> <font color="Maroon" title="c1">n</font></span>)</span><br/>  for <font color="Olive" title="b3">p1</font>, <font color="Olive" title="b4">p2</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">n</font></span>)</span>  st <font color="Olive" title="b1">P1</font> <a href="topreal1.html#R1" title="TOPREAL1:pred.1">is_an_arc_of</a> <font color="Olive" title="b3">p1</font>,<font color="Olive" title="b4">p2</font> &amp; <font color="Olive" title="b2">P2</font> <a href="topreal1.html#R1" title="TOPREAL1:pred.1">is_an_arc_of</a> <font color="Olive" title="b3">p1</font>,<font color="Olive" title="b4">p2</font> &amp; <font color="Olive" title="b1">P1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b2">P2</font> holds <br/><font color="Olive" title="b1">P1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">P2</font></span><br/><span class="kw">let </span><font color="Maroon" title="c2">P1</font>, <font color="Maroon" title="c3">P2</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">n</font></span>)</span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">p1</font>, <font color="Olive" title="b2">p2</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">n</font></span>)</span>  st <font color="Maroon" title="c2">P1</font> <a href="topreal1.html#R1" title="TOPREAL1:pred.1">is_an_arc_of</a> <font color="Olive" title="b1">p1</font>,<font color="Olive" title="b2">p2</font> &amp; <font color="Maroon" title="c3">P2</font> <a href="topreal1.html#R1" title="TOPREAL1:pred.1">is_an_arc_of</a> <font color="Olive" title="b1">p1</font>,<font color="Olive" title="b2">p2</font> &amp; <font color="Maroon" title="c2">P1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">P2</font> holds <br/><font color="Maroon" title="c2">P1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">P2</font></span><br/><span class="kw">let </span><font color="Maroon" title="c4">p1</font>, <font color="Maroon" title="c5">p2</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">n</font></span>)</span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c2">P1</font> <a href="topreal1.html#R1" title="TOPREAL1:pred.1">is_an_arc_of</a> <font color="Maroon" title="c4">p1</font>,<font color="Maroon" title="c5">p2</font> &amp; <font color="Maroon" title="c3">P2</font> <a href="topreal1.html#R1" title="TOPREAL1:pred.1">is_an_arc_of</a> <font color="Maroon" title="c4">p1</font>,<font color="Maroon" title="c5">p2</font> &amp; <font color="Maroon" title="c2">P1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">P2</font> implies <font color="Maroon" title="c2">P1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">P2</font> )</span><br/>





<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:49"/><span class="lab"><font color="Green" title="E31">A1</font></span>: 
<font color="Maroon" title="c2">P1</font> <a href="topreal1.html#R1" title="TOPREAL1:pred.1">is_an_arc_of</a> <font color="Maroon" title="c4">p1</font>,<font color="Maroon" title="c5">p2</font>
 <span class="kw">and </span><br/><a NAME="E2:49"/><span class="lab"><font color="Green" title="E32">A2</font></span>: 
<font color="Maroon" title="c3">P2</font> <a href="topreal1.html#R1" title="TOPREAL1:pred.1">is_an_arc_of</a> <font color="Maroon" title="c4">p1</font>,<font color="Maroon" title="c5">p2</font>
 <span class="kw">and </span><br/><a NAME="E3:49"/><span class="lab"><font color="Green" title="E33">A3</font></span>: 
<font color="Maroon" title="c2">P1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">P2</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c2">P1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">P2</font></span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c6">P19</font> = <font color="Maroon" title="c2">P1</font>, <font color="Maroon" title="c7">P29</font> = <font color="Maroon" title="c3">P2</font> as   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">n</font></span>)</span> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:49"><span class="lab"><font color="Green" title="E31">A1</font></span></a>, <a class="txt" href="#E2:49"><span class="lab"><font color="Green" title="E32">A2</font></span></a>, <a class="ref" href="topreal1.html#T1" onmouseover="rs('topreal1/T1')" onmouseout="rh()">TOPREAL1:1</a></span>;<br/>
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="49_1"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  not <font color="Maroon" title="c3">P2</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c2">P1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:49_1"/><span class="lab"><font color="Green" title="E34">A4</font></span>: 
<font color="Maroon" title="c3">P2</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c2">P1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><span class="kw">set </span><font color="Maroon" title="c8">p</font> =  the    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c3">P2</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c2">P1</font>;<br/><a NAME="E2:49_1"/><span class="lab"><font color="Green" title="E35">A5</font></span>: 
 the    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c3">P2</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c2">P1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c3">P2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:49_1"><span class="lab"><font color="Green" title="E34">A4</font></span></a>, <a class="ref" href="xboole_0.html#D5" onmouseover="rs('xboole_0/D5')" onmouseout="rh()">XBOOLE_0:def 5</a></span>;<br/><a NAME="E3:49_1"/><span class="lab"><font color="Green" title="E36">A6</font></span>: 
 not  the    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c3">P2</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c2">P1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c2">P1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:49_1"><span class="lab"><font color="Green" title="E34">A4</font></span></a>, <a class="ref" href="xboole_0.html#D5" onmouseover="rs('xboole_0/D5')" onmouseout="rh()">XBOOLE_0:def 5</a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c9">f1</font> being   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <a href="borsuk_1.html#K17" title="BORSUK_1:func.17">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> <font color="Maroon" title="c1">n</font></span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c6">P19</font></span>)</span><span class="kw"> such that </span><br/><a NAME="E5:49_1"/><span class="lab"><font color="Green" title="E37">A7</font></span>: 
<font color="Maroon" title="c9">f1</font> is  <a href="tops_2.html#V3" title="TOPS_2:attr.3">being_homeomorphism</a> 
 <span class="kw">and </span><br/><a NAME="E6:49_1"/><span class="lab"><font color="Green" title="E38">A8</font></span>: 
<font color="Maroon" title="c9">f1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">p1</font>
 <span class="kw">and </span><br/><a NAME="E7:49_1"/><span class="lab"><font color="Green" title="E39">A9</font></span>: 
<font color="Maroon" title="c9">f1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">p2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:49"><span class="lab"><font color="Green" title="E31">A1</font></span></a>, <a class="ref" href="topreal1.html#D1" onmouseover="rs('topreal1/D1')" onmouseout="rh()">TOPREAL1:def 1</a></span>;<br/><a NAME="E8:49_1"/><span class="lab"><font color="Green" title="E40">A10</font></span>: 
<font color="Maroon" title="c9">f1</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="#E5:49_1"><span class="lab"><font color="Green" title="E37">A7</font></span></a>, <a class="ref" href="tops_2.html#D5" onmouseover="rs('tops_2/D5')" onmouseout="rh()">TOPS_2:def 5</a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c10">f2</font> being   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <a href="borsuk_1.html#K17" title="BORSUK_1:func.17">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> <font color="Maroon" title="c1">n</font></span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c7">P29</font></span>)</span><span class="kw"> such that </span><br/><a NAME="E10:49_1"/><span class="lab"><font color="Green" title="E41">A11</font></span>: 
<font color="Maroon" title="c10">f2</font> is  <a href="tops_2.html#V3" title="TOPS_2:attr.3">being_homeomorphism</a> 
 <span class="kw">and </span><br/><a NAME="E11:49_1"/><span class="lab"><font color="Green" title="E42">A12</font></span>: 
<font color="Maroon" title="c10">f2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">p1</font>
 <span class="kw">and </span><br/><a NAME="E12:49_1"/><span class="lab"><font color="Green" title="E43">A13</font></span>: 
<font color="Maroon" title="c10">f2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">p2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:49"><span class="lab"><font color="Green" title="E32">A2</font></span></a>, <a class="ref" href="topreal1.html#D1" onmouseover="rs('topreal1/D1')" onmouseout="rh()">TOPREAL1:def 1</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c11">f4</font> = <font color="Maroon" title="c10">f2</font> as   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a> ;<br/><a NAME="E14:49_1"/><span class="lab"><font color="Green" title="E44">A14</font></span>: 
<font color="Maroon" title="c10">f2</font> is  <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:49_1"><span class="lab"><font color="Green" title="E41">A11</font></span></a>, <a class="ref" href="tops_2.html#D5" onmouseover="rs('tops_2/D5')" onmouseout="rh()">TOPS_2:def 5</a></span>;<br/><a NAME="E15:49_1"/><span class="lab"><font color="Green" title="E45">A15</font></span>: 
 <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c10">f2</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> <font color="Maroon" title="c1">n</font></span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c3">P2</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:49_1"><span class="lab"><font color="Green" title="E41">A11</font></span></a>, <a class="ref" href="tops_2.html#D5" onmouseover="rs('tops_2/D5')" onmouseout="rh()">TOPS_2:def 5</a></span>;<br/><a NAME="E16:49_1"/><span class="lab"><font color="Green" title="E46">A16</font></span>: 
<font color="Maroon" title="c10">f2</font> <a href="tops_2.html#NK3" title="TOPS_2:NK.3">"</a>  is  <a href="tops_2.html#V3" title="TOPS_2:attr.3">being_homeomorphism</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:49_1"><span class="lab"><font color="Green" title="E41">A11</font></span></a>, <a class="ref" href="tops_2.html#T56" onmouseover="rs('tops_2/T56')" onmouseout="rh()">TOPS_2:56</a></span>;<br/><span class="kw">then </span><span class="lab"><font color="Green" title="E47">A17</font></span>:  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">f2</font> <a href="tops_2.html#NK3" title="TOPS_2:NK.3">"</a></span>)</span> = 
 <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c3">P2</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="tops_2.html#D5" onmouseover="rs('tops_2/D5')" onmouseout="rh()">TOPS_2:def 5</a></span>
<br/>.= 
<font color="Maroon" title="c3">P2</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="pre_topc.html#D5" onmouseover="rs('pre_topc/D5')" onmouseout="rh()">PRE_TOPC:def 5</a></span>
;<br/><a NAME="E18:49_1"/>
<font color="Maroon" title="c10">f2</font> <a href="tops_2.html#NK3" title="TOPS_2:NK.3">"</a>  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="#E10:49_1"><span class="lab"><font color="Green" title="E41">A11</font></span></a>, <a class="ref" href="tops_2.html#D5" onmouseover="rs('tops_2/D5')" onmouseout="rh()">TOPS_2:def 5</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c12">h</font> being   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <a href="borsuk_1.html#K17" title="BORSUK_1:func.17">I[01]</a>,<a href="borsuk_1.html#K17" title="BORSUK_1:func.17">I[01]</a><span class="kw"> such that </span><br/><a NAME="E20:49_1"/><span class="lab"><font color="Green" title="E48">A18</font></span>: 
<font color="Maroon" title="c12">h</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">f2</font> <a href="tops_2.html#NK3" title="TOPS_2:NK.3">"</a></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c9">f1</font>
 <span class="kw">and </span><br/><a NAME="E21:49_1"/><span class="lab"><font color="Green" title="E49">A19</font></span>: 
<font color="Maroon" title="c12">h</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="#E3:49"><span class="lab"><font color="Green" title="E33">A3</font></span></a>, <a class="txt" href="#E8:49_1"><span class="lab"><font color="Green" title="E40">A10</font></span></a>, <a class="ref" href="jordan6.html#T45" target="_self" onmouseover="rs('jordan6/T45')" onmouseout="rh()">Th45</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c13">h1</font> = <font color="Maroon" title="c12">h</font> as   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K4" title="TOPMETR:func.4">Closed-Interval-TSpace</a> (<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,1)</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="borsuk_1.html#T40" onmouseover="rs('borsuk_1/T40')" onmouseout="rh()">BORSUK_1:40</a>, <a class="ref" href="funct_2.html#T7" onmouseover="rs('funct_2/T7')" onmouseout="rh()">FUNCT_2:7</a>, <a class="ref" href="topmetr.html#T17" onmouseover="rs('topmetr/T17')" onmouseout="rh()">TOPMETR:17</a>, <a class="ref" href="topmetr.html#T20" onmouseover="rs('topmetr/T20')" onmouseout="rh()">TOPMETR:20</a></span>;<br/><a NAME="E23:49_1"/>
 for <font color="Olive" title="b1">F1</font> being   <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>  st <font color="Olive" title="b1">F1</font> is  <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a>  holds <br/><font color="Maroon" title="c13">h1</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</a> <font color="Olive" title="b1">F1</font> is  <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a> 
 <div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="49_1_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c14">F1</font> be   <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>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c14">F1</font> is  <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a>  implies <font color="Maroon" title="c13">h1</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</a> <font color="Maroon" title="c14">F1</font> is  <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a>  )</span><br/>

<span class="kw">assume </span><a NAME="E1:49_1_2"/><span class="lab"><font color="Green" title="E50">A20</font></span>: 
<font color="Maroon" title="c14">F1</font> is  <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a> 
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c13">h1</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</a> <font color="Maroon" title="c14">F1</font> is  <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a> </span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c15">K</font> =  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <a href="borsuk_1.html#K17" title="BORSUK_1:func.17">I[01]</a> 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="borsuk_1.html#T40" onmouseover="rs('borsuk_1/T40')" onmouseout="rh()">BORSUK_1:40</a>, <a class="ref" href="topmetr.html#T17" onmouseover="rs('topmetr/T17')" onmouseout="rh()">TOPMETR:17</a></span>;<br/>
<a NAME="E3:49_1_2"/><span class="lab"><font color="Green" title="E51">A21</font></span>: 
 <a href="borsuk_1.html#K17" title="BORSUK_1:func.17">I[01]</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="topmetr.html#K3" title="TOPMETR:func.3">R^1</a> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c15">K</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="borsuk_1.html#T40" onmouseover="rs('borsuk_1/T40')" onmouseout="rh()">BORSUK_1:40</a>, <a class="ref" href="topmetr.html#T19" onmouseover="rs('topmetr/T19')" onmouseout="rh()">TOPMETR:19</a>, <a class="ref" href="topmetr.html#T20" onmouseover="rs('topmetr/T20')" onmouseout="rh()">TOPMETR:20</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c16">P3</font> = <font color="Maroon" title="c14">F1</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c15">K</font> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K3" title="TOPMETR:func.3">R^1</a> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c15">K</font></span>)</span> <span class="kw">by</span> <span class="lab"><a class="ref" href="tops_2.html#T29" onmouseover="rs('tops_2/T29')" onmouseout="rh()">TOPS_2:29</a></span>;<br/>
<a NAME="E5:49_1_2"/><span class="lab"><font color="Green" title="E52">A22</font></span>: 
<font color="Maroon" title="c16">P3</font> is  <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:49_1_2"><span class="lab"><font color="Green" title="E50">A20</font></span></a>, <a class="ref" href="jordan6.html#T2" target="_self" onmouseover="rs('jordan6/T2')" onmouseout="rh()">Th2</a></span>;<br/>
<font color="Maroon" title="c13">h1</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</a> <font color="Maroon" title="c15">K</font> = 
 the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <a href="borsuk_1.html#K17" title="BORSUK_1:func.17">I[01]</a>
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#T40" onmouseover="rs('funct_2/T40')" onmouseout="rh()">FUNCT_2:40</a></span>
<br/>.= 
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c13">h1</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D1" onmouseover="rs('funct_2/D1')" onmouseout="rh()">FUNCT_2:def 1</a></span>
;<br/>
<span class="kw">then </span><font color="Maroon" title="c13">h1</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</a> <font color="Maroon" title="c14">F1</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c13">h1</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</a> <font color="Maroon" title="c14">F1</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">h1</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</a> <font color="Maroon" title="c15">K</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#T132" onmouseover="rs('relat_1/T132')" onmouseout="rh()">RELAT_1:132</a>, <a class="ref" href="xboole_1.html#T28" onmouseover="rs('xboole_1/T28')" onmouseout="rh()">XBOOLE_1:28</a></span>
<br/>.= 
<font color="Maroon" title="c12">h</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</a> <font color="Maroon" title="c16">P3</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#T68" onmouseover="rs('funct_1/T68')" onmouseout="rh()">FUNCT_1:68</a></span>
;<br/>
<span class="kw">hence </span><a NAME="E8:49_1_2"/>
<font color="Maroon" title="c13">h1</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</a> <font color="Maroon" title="c14">F1</font> is  <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E21:49_1"><span class="lab"><font color="Green" title="E49">A19</font></span></a>, <a class="txt" href="#E3:49_1_2"><span class="lab"><font color="Green" title="E51">A21</font></span></a>, <a class="txt" href="#E5:49_1_2"><span class="lab"><font color="Green" title="E52">A22</font></span></a>, <a class="ref" href="topmetr.html#T20" onmouseover="rs('topmetr/T20')" onmouseout="rh()">TOPMETR:20</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c14">g</font> = <font color="Maroon" title="c13">h1</font> as   <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a>  <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K4" title="TOPMETR:func.4">Closed-Interval-TSpace</a> (<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,1)</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#D6" onmouseover="rs('pre_topc/D6')" onmouseout="rh()">PRE_TOPC:def 6</a></span>;<br/><span class="lab"><font color="Green" title="E50">A23</font></span>:  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c9">f1</font> = 
 <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <a href="borsuk_1.html#K17" title="BORSUK_1:func.17">I[01]</a>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:49_1"><span class="lab"><font color="Green" title="E37">A7</font></span></a>, <a class="ref" href="tops_2.html#D5" onmouseover="rs('tops_2/D5')" onmouseout="rh()">TOPS_2:def 5</a></span>
<br/>.= 
<span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">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#T40" onmouseover="rs('borsuk_1/T40')" onmouseout="rh()">BORSUK_1:40</a></span>
;<br/><a NAME="E26:49_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E51">A24</font></span>: 
 <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c9">f1</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_1.html#T1" onmouseover="rs('xxreal_1/T1')" onmouseout="rh()">XXREAL_1:1</a></span>;<br/><a NAME="E27:49_1"/><span class="lab"><font color="Green" title="E52">A25</font></span>: 
1 <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c9">f1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E25:49_1"><span class="lab"><font color="Green" title="E50">A23</font></span></a>, <a class="ref" href="xxreal_1.html#T1" onmouseover="rs('xxreal_1/T1')" onmouseout="rh()">XXREAL_1:1</a></span>;<br/><a NAME="E28:49_1"/><span class="lab"><font color="Green" title="E53">A26</font></span>: 
<font color="Maroon" title="c14">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">f2</font> <a href="tops_2.html#NK3" title="TOPS_2:NK.3">"</a></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">p1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:49_1"><span class="lab"><font color="Green" title="E38">A8</font></span></a>, <a class="txt" href="#E20:49_1"><span class="lab"><font color="Green" title="E48">A18</font></span></a>, <a class="txt" href="#E26:49_1"><span class="lab"><font color="Green" title="E51">A24</font></span></a>, <a class="ref" href="funct_1.html#T13" onmouseover="rs('funct_1/T13')" onmouseout="rh()">FUNCT_1:13</a></span>;<br/><a NAME="E29:49_1"/><span class="lab"><font color="Green" title="E54">A27</font></span>: 
<font color="Maroon" title="c14">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">f2</font> <a href="tops_2.html#NK3" title="TOPS_2:NK.3">"</a></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">p2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:49_1"><span class="lab"><font color="Green" title="E39">A9</font></span></a>, <a class="txt" href="#E20:49_1"><span class="lab"><font color="Green" title="E48">A18</font></span></a>, <a class="txt" href="#E27:49_1"><span class="lab"><font color="Green" title="E52">A25</font></span></a>, <a class="ref" href="funct_1.html#T13" onmouseover="rs('funct_1/T13')" onmouseout="rh()">FUNCT_1:13</a></span>;<br/><span class="lab"><font color="Green" title="E55">A28</font></span>:  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c10">f2</font> = 
 <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <a href="borsuk_1.html#K17" title="BORSUK_1:func.17">I[01]</a>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:49_1"><span class="lab"><font color="Green" title="E41">A11</font></span></a>, <a class="ref" href="tops_2.html#D5" onmouseover="rs('tops_2/D5')" onmouseout="rh()">TOPS_2:def 5</a></span>
<br/>.= 
<span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">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#T40" onmouseover="rs('borsuk_1/T40')" onmouseout="rh()">BORSUK_1:40</a></span>
;<br/><a NAME="E31:49_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E56">A29</font></span>: 
 <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c10">f2</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_1.html#T1" onmouseover="rs('xxreal_1/T1')" onmouseout="rh()">XXREAL_1:1</a></span>;<br/><a NAME="E32:49_1"/><span class="lab"><font color="Green" title="E57">A30</font></span>: 
1 <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c10">f2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E30:49_1"><span class="lab"><font color="Green" title="E55">A28</font></span></a>, <a class="ref" href="xxreal_1.html#T1" onmouseover="rs('xxreal_1/T1')" onmouseout="rh()">XXREAL_1:1</a></span>;<br/><a NAME="E33:49_1"/><span class="lab"><font color="Green" title="E58">A31</font></span>: 
<font color="Maroon" title="c10">f2</font> is  <a href="funct_2.html#V2" title="FUNCT_2:attr.2">onto</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:49_1"><span class="lab"><font color="Green" title="E45">A15</font></span></a>, <a class="ref" href="funct_2.html#D3" onmouseover="rs('funct_2/D3')" onmouseout="rh()">FUNCT_2:def 3</a></span>;<br/><a NAME="E34:49_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E59">A32</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c10">f2</font> <a href="tops_2.html#NK3" title="TOPS_2:NK.3">"</a></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">p1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">f4</font> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">p1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:49_1"><span class="lab"><font color="Green" title="E44">A14</font></span></a>, <a class="ref" href="tops_2.html#D4" onmouseover="rs('tops_2/D4')" onmouseout="rh()">TOPS_2:def 4</a></span>;<br/><a NAME="E35:49_1"/><span class="lab"><font color="Green" title="E60">A33</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c10">f2</font> <a href="tops_2.html#NK3" title="TOPS_2:NK.3">"</a></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">p2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">f4</font> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">p2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:49_1"><span class="lab"><font color="Green" title="E44">A14</font></span></a>, <a class="txt" href="#E33:49_1"><span class="lab"><font color="Green" title="E58">A31</font></span></a>, <a class="ref" href="tops_2.html#D4" onmouseover="rs('tops_2/D4')" onmouseout="rh()">TOPS_2:def 4</a></span>;<br/><a NAME="E36:49_1"/><span class="lab"><font color="Green" title="E61">A34</font></span>: 
<font color="Maroon" title="c14">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:49_1"><span class="lab"><font color="Green" title="E42">A12</font></span></a>, <a class="txt" href="#E14:49_1"><span class="lab"><font color="Green" title="E44">A14</font></span></a>, <a class="txt" href="#E28:49_1"><span class="lab"><font color="Green" title="E53">A26</font></span></a>, <a class="txt" href="#E31:49_1"><span class="lab"><font color="Green" title="E56">A29</font></span></a>, <a class="txt" href="#E34:49_1"><span class="lab"><font color="Green" title="E59">A32</font></span></a>, <a class="ref" href="funct_1.html#T32" onmouseover="rs('funct_1/T32')" onmouseout="rh()">FUNCT_1:32</a></span>;<br/><a NAME="E37:49_1"/><span class="lab"><font color="Green" title="E62">A35</font></span>: 
<font color="Maroon" title="c14">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:49_1"><span class="lab"><font color="Green" title="E43">A13</font></span></a>, <a class="txt" href="#E14:49_1"><span class="lab"><font color="Green" title="E44">A14</font></span></a>, <a class="txt" href="#E29:49_1"><span class="lab"><font color="Green" title="E54">A27</font></span></a>, <a class="txt" href="#E32:49_1"><span class="lab"><font color="Green" title="E57">A30</font></span></a>, <a class="txt" href="#E35:49_1"><span class="lab"><font color="Green" title="E60">A33</font></span></a>, <a class="ref" href="funct_1.html#T32" onmouseover="rs('funct_1/T32')" onmouseout="rh()">FUNCT_1:32</a></span>;<br/><a NAME="E38:49_1"/>
 the    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c3">P2</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c2">P1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c7">P29</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:49_1"><span class="lab"><font color="Green" title="E35">A5</font></span></a>, <a class="ref" href="pre_topc.html#T8" onmouseover="rs('pre_topc/T8')" onmouseout="rh()">PRE_TOPC:8</a></span>;<br/><a NAME="E39:49_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E63">A36</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c10">f2</font> <a href="tops_2.html#NK3" title="TOPS_2:NK.3">"</a></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a>  the    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c3">P2</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c2">P1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <a href="borsuk_1.html#K17" title="BORSUK_1:func.17">I[01]</a>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#T5" onmouseover="rs('funct_2/T5')" onmouseout="rh()">FUNCT_2:5</a></span>;<br/><div><span class="lab"><font color="Green" title="E64">A37</font></span>: <a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="49_1_5"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  not <span class="p1">(<span class="default"><font color="Maroon" title="c10">f2</font> <a href="tops_2.html#NK3" title="TOPS_2:NK.3">"</a></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a>  the    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c3">P2</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c2">P1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c14">g</font></span></a><div class="add"><span class="kw">assume </span><a NAME="E1:49_1_5"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c10">f2</font> <a href="tops_2.html#NK3" title="TOPS_2:NK.3">"</a></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a>  the    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c3">P2</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c2">P1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c14">g</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c15">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> <span class="kw"> such that </span><br/><a NAME="E3:49_1_5"/><span class="lab"><font color="Green" title="E65">A38</font></span>: 
<font color="Maroon" title="c15">x</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="c14">g</font>
 <span class="kw">and </span><br/><a NAME="E4:49_1_5"/><span class="lab"><font color="Green" title="E66">A39</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c10">f2</font> <a href="tops_2.html#NK3" title="TOPS_2:NK.3">"</a></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a>  the    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c3">P2</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c2">P1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">x</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#D3" onmouseover="rs('funct_1/D3')" onmouseout="rh()">FUNCT_1:def 3</a></span>;<br/><a NAME="E5:49_1_5"/><span class="lab"><font color="Green" title="E67">A40</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c10">f2</font> <a href="tops_2.html#NK3" title="TOPS_2:NK.3">"</a></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a>  the    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c3">P2</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c2">P1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">f2</font> <a href="tops_2.html#NK3" title="TOPS_2:NK.3">"</a></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">f1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">x</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:49_1"><span class="lab"><font color="Green" title="E48">A18</font></span></a>, <a class="txt" href="#E3:49_1_5"><span class="lab"><font color="Green" title="E65">A38</font></span></a>, <a class="txt" href="#E4:49_1_5"><span class="lab"><font color="Green" title="E66">A39</font></span></a>, <a class="ref" href="funct_1.html#T12" onmouseover="rs('funct_1/T12')" onmouseout="rh()">FUNCT_1:12</a></span>;<br/><a NAME="E6:49_1_5"/><span class="lab"><font color="Green" title="E68">A41</font></span>: 
<font color="Maroon" title="c15">x</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="c9">f1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:49_1"><span class="lab"><font color="Green" title="E48">A18</font></span></a>, <a class="txt" href="#E3:49_1_5"><span class="lab"><font color="Green" title="E65">A38</font></span></a>, <a class="ref" href="funct_1.html#T11" onmouseover="rs('funct_1/T11')" onmouseout="rh()">FUNCT_1:11</a></span>;<br/><a NAME="E7:49_1_5"/><span class="lab"><font color="Green" title="E69">A42</font></span>: 
<font color="Maroon" title="c9">f1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">f2</font> <a href="tops_2.html#NK3" title="TOPS_2:NK.3">"</a></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:49_1"><span class="lab"><font color="Green" title="E48">A18</font></span></a>, <a class="txt" href="#E3:49_1_5"><span class="lab"><font color="Green" title="E65">A38</font></span></a>, <a class="ref" href="funct_1.html#T11" onmouseover="rs('funct_1/T11')" onmouseout="rh()">FUNCT_1:11</a></span>;<br/><a NAME="E8:49_1_5"/>
<font color="Maroon" title="c10">f2</font> <a href="tops_2.html#NK3" title="TOPS_2:NK.3">"</a>  is  <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:49_1"><span class="lab"><font color="Green" title="E46">A16</font></span></a>, <a class="ref" href="tops_2.html#D5" onmouseover="rs('tops_2/D5')" onmouseout="rh()">TOPS_2:def 5</a></span>;<br/><a NAME="E9:49_1_5"/><span class="kw">then </span>
 the    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c3">P2</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c2">P1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">f1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">x</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:49_1"><span class="lab"><font color="Green" title="E35">A5</font></span></a>, <a class="txt" href="#E17:49_1"><span class="lab"><font color="Green" title="E47">A17</font></span></a>, <a class="txt" href="#E5:49_1_5"><span class="lab"><font color="Green" title="E67">A40</font></span></a>, <a class="txt" href="#E7:49_1_5"><span class="lab"><font color="Green" title="E69">A42</font></span></a>, <a class="ref" href="funct_1.html#D4" onmouseover="rs('funct_1/D4')" onmouseout="rh()">FUNCT_1:def 4</a></span>;<br/><a NAME="E10:49_1_5"/><span class="kw">then </span><span class="lab"><font color="Green" title="E70">A43</font></span>: 
 the    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c3">P2</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c2">P1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c9">f1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:49_1_5"><span class="lab"><font color="Green" title="E68">A41</font></span></a>, <a class="ref" href="funct_1.html#D3" onmouseover="rs('funct_1/D3')" onmouseout="rh()">FUNCT_1:def 3</a></span>;<br/> <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c9">f1</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> <font color="Maroon" title="c1">n</font></span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c2">P1</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:49_1"><span class="lab"><font color="Green" title="E37">A7</font></span></a>, <a class="ref" href="tops_2.html#D5" onmouseover="rs('tops_2/D5')" onmouseout="rh()">TOPS_2:def 5</a></span>
<br/>.= 
<font color="Maroon" title="c2">P1</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="pre_topc.html#D5" onmouseover="rs('pre_topc/D5')" onmouseout="rh()">PRE_TOPC:def 5</a></span>
;<br/><span class="kw">hence </span><a NAME="E12:49_1_5"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:49_1"><span class="lab"><font color="Green" title="E34">A4</font></span></a>, <a class="txt" href="#E10:49_1_5"><span class="lab"><font color="Green" title="E70">A43</font></span></a>, <a class="ref" href="xboole_0.html#D5" onmouseover="rs('xboole_0/D5')" onmouseout="rh()">XBOOLE_0:def 5</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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><span class="kw">reconsider </span><font color="Maroon" title="c15">r</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c10">f2</font> <a href="tops_2.html#NK3" title="TOPS_2:NK.3">"</a></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a>  the    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c3">P2</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c2">P1</font> as   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a> ;<br/><span class="lab"><font color="Green" title="E65">A44</font></span>:  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c10">f2</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> <font color="Maroon" title="c1">n</font></span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c3">P2</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:49_1"><span class="lab"><font color="Green" title="E41">A11</font></span></a>, <a class="ref" href="tops_2.html#D5" onmouseover="rs('tops_2/D5')" onmouseout="rh()">TOPS_2:def 5</a></span>
<br/>.= 
<font color="Maroon" title="c3">P2</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="pre_topc.html#D5" onmouseover="rs('pre_topc/D5')" onmouseout="rh()">PRE_TOPC:def 5</a></span>
;<br/><a NAME="E43:49_1"/><span class="lab"><font color="Green" title="E66">A45</font></span>: 
<font color="Maroon" title="c15">r</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="#E39:49_1"><span class="lab"><font color="Green" title="E63">A36</font></span></a>, <a class="ref" href="borsuk_1.html#T40" onmouseover="rs('borsuk_1/T40')" onmouseout="rh()">BORSUK_1:40</a>, <a class="ref" href="xxreal_1.html#T1" onmouseover="rs('xxreal_1/T1')" onmouseout="rh()">XXREAL_1:1</a></span>;<br/><div><span class="lab"><font color="Green" title="E67">A46</font></span>: <a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="49_1_7"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  not <font color="Maroon" title="c15">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:49_1_7"/><span class="lab"><font color="Green" title="E68">A47</font></span>: 
<font color="Maroon" title="c15">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> 
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><font color="Maroon" title="c10">f2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">r</font> = 
<font color="Maroon" title="c11">f4</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c11">f4</font> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a>  the    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c3">P2</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c2">P1</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E33:49_1"><span class="lab"><font color="Green" title="E58">A31</font></span></a>, <a class="txt" href="#E14:49_1"><span class="lab"><font color="Green" title="E44">A14</font></span></a>, <a class="ref" href="tops_2.html#D4" onmouseover="rs('tops_2/D4')" onmouseout="rh()">TOPS_2:def 4</a></span>
<br/>.= 
 the    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c3">P2</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c2">P1</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:49_1"><span class="lab"><font color="Green" title="E35">A5</font></span></a>, <a class="txt" href="#E14:49_1"><span class="lab"><font color="Green" title="E44">A14</font></span></a>, <a class="txt" href="#E42:49_1"><span class="lab"><font color="Green" title="E65">A44</font></span></a>, <a class="ref" href="funct_1.html#T35" onmouseover="rs('funct_1/T35')" onmouseout="rh()">FUNCT_1:35</a></span>
;<br/><span class="kw">hence </span><a NAME="E3:49_1_7"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:49"><span class="lab"><font color="Green" title="E31">A1</font></span></a>, <a class="txt" href="#E3:49_1"><span class="lab"><font color="Green" title="E36">A6</font></span></a>, <a class="txt" href="#E11:49_1"><span class="lab"><font color="Green" title="E42">A12</font></span></a>, <a class="txt" href="#E1:49_1_7"><span class="lab"><font color="Green" title="E68">A47</font></span></a>, <a class="ref" href="topreal1.html#T1" onmouseover="rs('topreal1/T1')" onmouseout="rh()">TOPREAL1:1</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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><div><span class="lab"><font color="Green" title="E68">A48</font></span>: <a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="49_1_8"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  not <font color="Maroon" title="c15">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1</span></a><div class="add"><span class="kw">assume </span><a NAME="E1:49_1_8"/><span class="lab"><font color="Green" title="E69">A49</font></span>: 
<font color="Maroon" title="c15">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><font color="Maroon" title="c10">f2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">r</font> = 
<font color="Maroon" title="c11">f4</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c11">f4</font> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a>  the    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c3">P2</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c2">P1</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E33:49_1"><span class="lab"><font color="Green" title="E58">A31</font></span></a>, <a class="txt" href="#E14:49_1"><span class="lab"><font color="Green" title="E44">A14</font></span></a>, <a class="ref" href="tops_2.html#D4" onmouseover="rs('tops_2/D4')" onmouseout="rh()">TOPS_2:def 4</a></span>
<br/>.= 
 the    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c3">P2</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c2">P1</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:49_1"><span class="lab"><font color="Green" title="E35">A5</font></span></a>, <a class="txt" href="#E14:49_1"><span class="lab"><font color="Green" title="E44">A14</font></span></a>, <a class="txt" href="#E42:49_1"><span class="lab"><font color="Green" title="E65">A44</font></span></a>, <a class="ref" href="funct_1.html#T35" onmouseover="rs('funct_1/T35')" onmouseout="rh()">FUNCT_1:35</a></span>
;<br/><span class="kw">hence </span><a NAME="E3:49_1_8"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:49"><span class="lab"><font color="Green" title="E31">A1</font></span></a>, <a class="txt" href="#E3:49_1"><span class="lab"><font color="Green" title="E36">A6</font></span></a>, <a class="txt" href="#E12:49_1"><span class="lab"><font color="Green" title="E43">A13</font></span></a>, <a class="txt" href="#E1:49_1_8"><span class="lab"><font color="Green" title="E69">A49</font></span></a>, <a class="ref" href="topreal1.html#T1" onmouseover="rs('topreal1/T1')" onmouseout="rh()">TOPREAL1:1</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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="E46:49_1"/><span class="lab"><font color="Green" title="E69">A50</font></span>: 
 <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c15">r</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E39:49_1"><span class="lab"><font color="Green" title="E63">A36</font></span></a>, <a class="txt" href="#E44:49_1"><span class="lab"><font color="Green" title="E67">A46</font></span></a>, <a class="ref" href="borsuk_1.html#T40" onmouseover="rs('borsuk_1/T40')" onmouseout="rh()">BORSUK_1:40</a>, <a class="ref" href="xxreal_1.html#T1" onmouseover="rs('xxreal_1/T1')" onmouseout="rh()">XXREAL_1:1</a></span>;<br/><a NAME="E47:49_1"/>
<font color="Maroon" title="c15">r</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E43:49_1"><span class="lab"><font color="Green" title="E66">A45</font></span></a>, <a class="txt" href="#E45:49_1"><span class="lab"><font color="Green" title="E68">A48</font></span></a>, <a class="ref" href="xxreal_0.html#T1" onmouseover="rs('xxreal_0/T1')" onmouseout="rh()">XXREAL_0:1</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c16">rc</font> being   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a><span class="kw"> such that </span><br/><a NAME="E49:49_1"/><span class="lab"><font color="Green" title="E70">A51</font></span>: 
<font color="Maroon" title="c14">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">rc</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">r</font>
 <span class="kw">and </span><br/><a NAME="E50:49_1"/><span class="lab"><font color="Green" title="E71">A52</font></span>: 
 <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c16">rc</font>
 <span class="kw">and </span><br/><a NAME="E51:49_1"/><span class="lab"><font color="Green" title="E72">A53</font></span>: 
<font color="Maroon" title="c16">rc</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E36:49_1"><span class="lab"><font color="Green" title="E61">A34</font></span></a>, <a class="txt" href="#E37:49_1"><span class="lab"><font color="Green" title="E62">A35</font></span></a>, <a class="txt" href="#E46:49_1"><span class="lab"><font color="Green" title="E69">A50</font></span></a>, <a class="ref" href="topreal5.html#T6" onmouseover="rs('topreal5/T6')" onmouseout="rh()">TOPREAL5:6</a></span>;<br/> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c2">P1</font></span>)</span> = 
 <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c2">P1</font></span>)</span>

<br/>.= 
<font color="Maroon" title="c2">P1</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="pre_topc.html#D5" onmouseover="rs('pre_topc/D5')" onmouseout="rh()">PRE_TOPC:def 5</a></span>
;<br/><a NAME="E53:49_1"/><span class="kw">then </span>
 <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c9">f1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">f2</font> <a href="tops_2.html#NK3" title="TOPS_2:NK.3">"</a></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:49"><span class="lab"><font color="Green" title="E33">A3</font></span></a>, <a class="txt" href="#E17:49_1"><span class="lab"><font color="Green" title="E47">A17</font></span></a></span>;<br/><span class="kw">then </span> <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c14">g</font> = 
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c9">f1</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:49_1"><span class="lab"><font color="Green" title="E48">A18</font></span></a>, <a class="ref" href="relat_1.html#T27" onmouseover="rs('relat_1/T27')" onmouseout="rh()">RELAT_1:27</a></span>
<br/>.= 
 <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <a href="borsuk_1.html#K17" title="BORSUK_1:func.17">I[01]</a>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:49_1"><span class="lab"><font color="Green" title="E37">A7</font></span></a>, <a class="ref" href="tops_2.html#D5" onmouseover="rs('tops_2/D5')" onmouseout="rh()">TOPS_2:def 5</a></span>
<br/>.= 
<span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">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#T40" onmouseover="rs('borsuk_1/T40')" onmouseout="rh()">BORSUK_1:40</a></span>
;<br/><a NAME="E55:49_1"/><span class="kw">then </span>
<font color="Maroon" title="c16">rc</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="c14">g</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E50:49_1"><span class="lab"><font color="Green" title="E71">A52</font></span></a>, <a class="txt" href="#E51:49_1"><span class="lab"><font color="Green" title="E72">A53</font></span></a>, <a class="ref" href="xxreal_1.html#T1" onmouseover="rs('xxreal_1/T1')" onmouseout="rh()">XXREAL_1:1</a></span>;<br/><span class="kw">hence </span><a NAME="E56:49_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E40:49_1"><span class="lab"><font color="Green" title="E64">A37</font></span></a>, <a class="txt" href="#E49:49_1"><span class="lab"><font color="Green" title="E70">A51</font></span></a>, <a class="ref" href="funct_1.html#D3" onmouseover="rs('funct_1/D3')" onmouseout="rh()">FUNCT_1:def 3</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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="E6:49"/><span class="kw">then </span>
<font color="Maroon" title="c3">P2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">P1</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T37" onmouseover="rs('xboole_1/T37')" onmouseout="rh()">XBOOLE_1:37</a></span>;<br/>
<span class="kw">hence </span><a NAME="E7:49"/>
<font color="Maroon" title="c2">P1</font> <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="#E3:49"><span class="lab"><font color="Green" title="E33">A3</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
