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<b>let </b><font color="Maroon" title="c8">r</font> be  <a href="xxreal_0.html#V2" title="XXREAL_0:attr.2">positive</a> <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">o</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span><br/> for <font color="Olive" title="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="toprealb.html#K7" title="TOPREALB:func.7">Tcircle</a> <font color="Olive" title="b1">o</font>,<font color="Maroon" title="c8">r</font></span>)</span> holds   <a href="gr_cy_1.html#K2" title="GR_CY_1:func.2">INT.Group</a> , <a href="topalg_1.html#NK4" title="TOPALG_1:NK.4">pi_1</a> <span class="p1">(<span class="default"><a href="toprealb.html#K7" title="TOPREALB:func.7">Tcircle</a> <font color="Olive" title="b1">o</font>,<font color="Maroon" title="c8">r</font></span>)</span>,<font color="Olive" title="b2">x</font> <a href="group_6.html#R2" title="GROUP_6:pred.2">are_isomorphic</a> </span><br/><b>let </b><font color="Maroon" title="c9">o</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> 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="toprealb.html#K7" title="TOPREALB:func.7">Tcircle</a> <font color="Maroon" title="c9">o</font>,<font color="Maroon" title="c8">r</font></span>)</span> holds   <a href="gr_cy_1.html#K2" title="GR_CY_1:func.2">INT.Group</a> , <a href="topalg_1.html#NK4" title="TOPALG_1:NK.4">pi_1</a> <span class="p1">(<span class="default"><a href="toprealb.html#K7" title="TOPREALB:func.7">Tcircle</a> <font color="Maroon" title="c9">o</font>,<font color="Maroon" title="c8">r</font></span>)</span>,<font color="Olive" title="b1">x</font> <a href="group_6.html#R2" title="GROUP_6:pred.2">are_isomorphic</a> </span><br/><b>let </b><font color="Maroon" title="c10">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="toprealb.html#K7" title="TOPREALB:func.7">Tcircle</a> <font color="Maroon" title="c9">o</font>,<font color="Maroon" title="c8">r</font></span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="gr_cy_1.html#K2" title="GR_CY_1:func.2">INT.Group</a> , <a href="topalg_1.html#NK4" title="TOPALG_1:NK.4">pi_1</a> <span class="p1">(<span class="default"><a href="toprealb.html#K7" title="TOPREALB:func.7">Tcircle</a> <font color="Maroon" title="c9">o</font>,<font color="Maroon" title="c8">r</font></span>)</span>,<font color="Maroon" title="c10">x</font> <a href="group_6.html#R2" title="GROUP_6:pred.2">are_isomorphic</a> </span><br/>





<a NAME="E1:47"/>
 <a href="toprealb.html#K8" title="TOPREALB:func.8">Tunit_circle</a> 2 <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="euclid.html#K16" title="EUCLID:func.16">0.REAL</a> 2</span>)</span>,1
 
<b>by </b><i><a class="ref" href="toprealb.html#D7" title="TOPREALB:def.7">TOPREALB:def 7</a></i>;<br/>
<a NAME="E2:47"/><b>then </b>
 <a href="topalg_1.html#NK4" title="TOPALG_1:NK.4">pi_1</a> <span class="p1">(<span class="default"><a href="toprealb.html#K8" title="TOPREALB:func.8">Tunit_circle</a> 2</span>)</span>,<a href="toprealb.html#K9" title="TOPREALB:func.9">c[10]</a> , <a href="topalg_1.html#NK4" title="TOPALG_1:NK.4">pi_1</a> <span class="p1">(<span class="default"><a href="toprealb.html#K7" title="TOPREALB:func.7">Tcircle</a> <font color="Maroon" title="c9">o</font>,<font color="Maroon" title="c8">r</font></span>)</span>,<font color="Maroon" title="c10">x</font> <a href="group_6.html#R2" title="GROUP_6:pred.2">are_isomorphic</a> 
 
<b>by </b><i><a class="ref" href="topalg_3.html#T35" title="TOPALG_3:th.35">TOPALG_3:35</a>, <a class="ref" href="toprealb.html#T20" title="TOPREALB:th.20">TOPREALB:20</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c11">h</font> being   <a href="group_6.html#NM2" title="GROUP_6:NM.2">Homomorphism</a> of <span class="p1">(<span class="default"><a href="topalg_1.html#NK4" title="TOPALG_1:NK.4">pi_1</a> <span class="p2">(<span class="default"><a href="toprealb.html#K8" title="TOPREALB:func.8">Tunit_circle</a> 2</span>)</span>,<a href="toprealb.html#K9" title="TOPREALB:func.9">c[10]</a> </span>)</span>,<span class="p1">(<span class="default"><a href="topalg_1.html#NK4" title="TOPALG_1:NK.4">pi_1</a> <span class="p2">(<span class="default"><a href="toprealb.html#K7" title="TOPREALB:func.7">Tcircle</a> <font color="Maroon" title="c9">o</font>,<font color="Maroon" title="c8">r</font></span>)</span>,<font color="Maroon" title="c10">x</font></span>)</span><b> such that </b><br/><a NAME="E4:47"/><i><font color="Green" title="E43">A1</font></i>: 
<font color="Maroon" title="c11">h</font> <a href="group_6.html#NR3" title="GROUP_6:NR.3">is_isomorphism</a> 
 <b>by </b><i><a class="ref" href="group_6.html#D15" title="GROUP_6:def.15">GROUP_6:def 15</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c11">h</font> <a href="group_6.html#K7" title="GROUP_6:func.7">*</a> <a href="topalg_5.html#K8" title="TOPALG_5:func.8">Ciso</a> 
; <i><font color="Red">:: according to </font></i><a class="ref" href="group_6.html#D15" title="GROUP_6:def.15">GROUP_6:def 15</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c11">h</font> <a href="group_6.html#K7" title="GROUP_6:func.7">*</a> <a href="topalg_5.html#K8" title="TOPALG_5:func.8">Ciso</a>  is <a href="group_6.html#V4" title="GROUP_6:attr.4">being_isomorphism</a></span><br/>

<b>thus </b><a NAME="E5:47"/>
<font color="Maroon" title="c11">h</font> <a href="group_6.html#K7" title="GROUP_6:func.7">*</a> <a href="topalg_5.html#K8" title="TOPALG_5:func.8">Ciso</a>  is <a href="group_6.html#V4" title="GROUP_6:attr.4">being_isomorphism</a>
 <b>by </b><i><a class="txt" href="topalg_5.html#E4:47"><i><font color="Green" title="E43">A1</font></i></a>, <a class="ref" href="topalg_5.html#T26" target="_self" title="TOPALG_5:th.26">Th26</a>, <a class="ref" href="group_6.html#T74" title="GROUP_6:th.74">GROUP_6:74</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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