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<span class="kw">let </span><font color="Maroon" title="c1">n</font> be   non  <a href="ordinal1.html#V8" title="ORDINAL1:attr.8">zero</a>  <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">r</font> being   <a href="xxreal_0.html#V2" title="XXREAL_0:attr.2">positive</a>  <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a><br/>  for <font color="Olive" title="b2">x</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="c1">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 3 holds <br/> <a href="toprealb.html#K7" title="TOPREALB:func.7">Tcircle</a> (<font color="Olive" title="b2">x</font>,<font color="Olive" title="b1">r</font>) is  <a href="topalg_6.html#V1" title="TOPALG_6:attr.1">having_trivial_Fundamental_Group</a> </span><br/><span class="kw">let </span><font color="Maroon" title="c2">r</font> be   <a href="xxreal_0.html#V2" title="XXREAL_0:attr.2">positive</a>  <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</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">x</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="c1">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 3 holds <br/> <a href="toprealb.html#K7" title="TOPREALB:func.7">Tcircle</a> (<font color="Olive" title="b1">x</font>,<font color="Maroon" title="c2">r</font>) is  <a href="topalg_6.html#V1" title="TOPALG_6:attr.1">having_trivial_Fundamental_Group</a> </span><br/><span class="kw">let </span><font color="Maroon" title="c3">x</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="c1">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 3 implies  <a href="toprealb.html#K7" title="TOPREALB:func.7">Tcircle</a> (<font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c2">r</font>) is  <a href="topalg_6.html#V1" title="TOPALG_6:attr.1">having_trivial_Fundamental_Group</a>  )</span><br/>





<span class="kw">assume </span><a NAME="E1:84"/><span class="lab"><font color="Green" title="E55">A1</font></span>: 
<font color="Maroon" title="c1">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 3
 ; <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">  <a href="toprealb.html#K7" title="TOPREALB:func.7">Tcircle</a> (<font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c2">r</font>) is  <a href="topalg_6.html#V1" title="TOPALG_6:attr.1">having_trivial_Fundamental_Group</a> </span><br/>

<a NAME="E2:84"/><span class="kw">then </span>
<font color="Maroon" title="c1">n</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 3 <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T9" onmouseover="rs('xreal_1/T9')" onmouseout="rh()">XREAL_1:9</a></span>;<br/>
<a NAME="E3:84"/><span class="kw">then </span>
 <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">n</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_0.html#T2" onmouseover="rs('xxreal_0/T2')" onmouseout="rh()">XXREAL_0:2</a></span>;<br/>
<a NAME="E4:84"/><span class="kw">then </span><span class="lab"><font color="Green" title="E56">A2</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_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="c1">n</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_0.html#D2" onmouseover="rs('xreal_0/D2')" onmouseout="rh()">XREAL_0:def 2</a></span>;<br/>
<a NAME="E5:84"/>
2 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3
 
;<br/>
<a NAME="E6:84"/><span class="kw">then </span>
2 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">n</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:84"><span class="lab"><font color="Green" title="E55">A1</font></span></a>, <a class="ref" href="nat_d.html#T49" onmouseover="rs('nat_d/T49')" onmouseout="rh()">NAT_D:49</a></span>;<br/>
<a NAME="E7:84"/><span class="kw">then </span><span class="lab"><font color="Green" title="E57">A3</font></span>: 
 <a href="mfold_2.html#K4" title="MFOLD_2:func.4">TUnitSphere</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> is  <a href="topalg_6.html#V1" title="TOPALG_6:attr.1">having_trivial_Fundamental_Group</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="topalg_6.html#T46" target="_self" onmouseover="rs('topalg_6/T46')" onmouseout="rh()">Th46</a></span>;<br/>
<a NAME="E8:84"/><span class="lab"><font color="Green" title="E58">A4</font></span>: 
 <a href="mfold_2.html#K4" title="MFOLD_2:func.4">TUnitSphere</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="toprealb.html#K8" title="TOPREALB:func.8">Tunit_circle</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="mfold_2.html#D4" onmouseover="rs('mfold_2/D4')" onmouseout="rh()">MFOLD_2:def 4</a></span>;<br/>
<a NAME="E9:84"/><span class="lab"><font color="Green" title="E59">A5</font></span>: 
 <a href="toprealb.html#K8" title="TOPREALB:func.8">Tunit_circle</a> <font color="Maroon" title="c1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="toprealb.html#K7" title="TOPREALB:func.7">Tcircle</a> (<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> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span>,1)
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="toprealb.html#D7" onmouseover="rs('toprealb/D7')" onmouseout="rh()">TOPREALB:def 7</a></span>;<br/>
<a NAME="E10:84"/>
<font color="Maroon" title="c1">n</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal1.html#D12" onmouseover="rs('ordinal1/D12')" onmouseout="rh()">ORDINAL1:def 12</a></span>;<br/>
<a NAME="E11:84"/><span class="kw">then </span>
 <a href="toprealb.html#K7" title="TOPREALB:func.7">Tcircle</a> (<font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c2">r</font>), <a href="toprealb.html#K7" title="TOPREALB:func.7">Tcircle</a> (<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> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span>,1) <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="toprealb.html#T20" onmouseover="rs('toprealb/T20')" onmouseout="rh()">TOPREALB:20</a></span>;<br/>
<span class="kw">hence </span><a NAME="E12:84"/>
 <a href="toprealb.html#K7" title="TOPREALB:func.7">Tcircle</a> (<font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c2">r</font>) is  <a href="topalg_6.html#V1" title="TOPALG_6:attr.1">having_trivial_Fundamental_Group</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:84"><span class="lab"><font color="Green" title="E56">A2</font></span></a>, <a class="txt" href="#E7:84"><span class="lab"><font color="Green" title="E57">A3</font></span></a>, <a class="txt" href="#E8:84"><span class="lab"><font color="Green" title="E58">A4</font></span></a>, <a class="txt" href="#E9:84"><span class="lab"><font color="Green" title="E59">A5</font></span></a>, <a class="ref" href="topalg_6.html#T13" target="_self" onmouseover="rs('topalg_6/T13')" onmouseout="rh()">Th13</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/>


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