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<span class="kw">let </span><font color="Maroon" title="c4">S</font> be  non  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>  <a href="pre_topc.html#NM1" title="PRE_TOPC:NM.1">TopSpace</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">T</font> being non  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>   <a href="borsuk_2.html#V1" title="BORSUK_2:attr.1">arcwise_connected</a>  <a href="pre_topc.html#NM1" title="PRE_TOPC:NM.1">TopSpace</a><br/> for <font color="Olive" title="b2">s</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c4">S</font><br/> for <font color="Olive" title="b3">t</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Olive" title="b1">T</font>  st <font color="Maroon" title="c4">S</font>,<font color="Olive" title="b1">T</font> <a href="borsuk_3.html#R2" title="BORSUK_3:pred.2">are_homeomorphic</a>  holds <br/> <a href="topalg_1.html#NK6" title="TOPALG_1:NK.6">pi_1</a> <font color="Maroon" title="c4">S</font>,<font color="Olive" title="b2">s</font>, <a href="topalg_1.html#NK6" title="TOPALG_1:NK.6">pi_1</a> <font color="Olive" title="b1">T</font>,<font color="Olive" title="b3">t</font> <a href="group_6.html#R2" title="GROUP_6:pred.2">are_isomorphic</a> </span><br/><span class="kw">let </span><font color="Maroon" title="c5">T</font> be  non  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>   <a href="borsuk_2.html#V1" title="BORSUK_2:attr.1">arcwise_connected</a>  <a href="pre_topc.html#NM1" title="PRE_TOPC:NM.1">TopSpace</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">s</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c4">S</font><br/> for <font color="Olive" title="b2">t</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c5">T</font>  st <font color="Maroon" title="c4">S</font>,<font color="Maroon" title="c5">T</font> <a href="borsuk_3.html#R2" title="BORSUK_3:pred.2">are_homeomorphic</a>  holds <br/> <a href="topalg_1.html#NK6" title="TOPALG_1:NK.6">pi_1</a> <font color="Maroon" title="c4">S</font>,<font color="Olive" title="b1">s</font>, <a href="topalg_1.html#NK6" title="TOPALG_1:NK.6">pi_1</a> <font color="Maroon" title="c5">T</font>,<font color="Olive" title="b2">t</font> <a href="group_6.html#R2" title="GROUP_6:pred.2">are_isomorphic</a> </span><br/><span class="kw">let </span><font color="Maroon" title="c6">s</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c4">S</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">t</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c5">T</font>  st <font color="Maroon" title="c4">S</font>,<font color="Maroon" title="c5">T</font> <a href="borsuk_3.html#R2" title="BORSUK_3:pred.2">are_homeomorphic</a>  holds <br/> <a href="topalg_1.html#NK6" title="TOPALG_1:NK.6">pi_1</a> <font color="Maroon" title="c4">S</font>,<font color="Maroon" title="c6">s</font>, <a href="topalg_1.html#NK6" title="TOPALG_1:NK.6">pi_1</a> <font color="Maroon" title="c5">T</font>,<font color="Olive" title="b1">t</font> <a href="group_6.html#R2" title="GROUP_6:pred.2">are_isomorphic</a> </span><br/><span class="kw">let </span><font color="Maroon" title="c7">t</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c5">T</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c4">S</font>,<font color="Maroon" title="c5">T</font> <a href="borsuk_3.html#R2" title="BORSUK_3:pred.2">are_homeomorphic</a>  implies  <a href="topalg_1.html#NK6" title="TOPALG_1:NK.6">pi_1</a> <font color="Maroon" title="c4">S</font>,<font color="Maroon" title="c6">s</font>, <a href="topalg_1.html#NK6" title="TOPALG_1:NK.6">pi_1</a> <font color="Maroon" title="c5">T</font>,<font color="Maroon" title="c7">t</font> <a href="group_6.html#R2" title="GROUP_6:pred.2">are_isomorphic</a>  )</span><br/>







<span class="kw">given </span><font color="Maroon" title="c8">f</font> being   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c4">S</font>,<font color="Maroon" title="c5">T</font><span class="kw"> such that </span><a NAME="E2:44"/><span class="lab"><font color="Green" title="E16">A1</font></span>: 
<font color="Maroon" title="c8">f</font> is  <a href="tops_2.html#V3" title="TOPS_2:attr.3">being_homeomorphism</a> 
 ; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="t_0topsp.html#D1" title="T_0TOPSP:def.1">T_0TOPSP:def 1</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="topalg_1.html#NK6" title="TOPALG_1:NK.6">pi_1</a> <font color="Maroon" title="c4">S</font>,<font color="Maroon" title="c6">s</font>, <a href="topalg_1.html#NK6" title="TOPALG_1:NK.6">pi_1</a> <font color="Maroon" title="c5">T</font>,<font color="Maroon" title="c7">t</font> <a href="group_6.html#R2" title="GROUP_6:pred.2">are_isomorphic</a> </span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c9">f</font> = <font color="Maroon" title="c8">f</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 <font color="Maroon" title="c4">S</font>,<font color="Maroon" title="c5">T</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:44"><span class="lab"><font color="Green" title="E16">A1</font></span></a>, <a class="ref" href="tops_2.html#D5" title="TOPS_2:def.5">TOPS_2:def 5</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c10">P</font> being    <a href="borsuk_2.html#M1" title="BORSUK_2:mode.1">Path</a> of <font color="Maroon" title="c7">t</font>,<font color="Maroon" title="c9">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">s</font>;<br/>
<span class="kw">take </span>
<span class="p1">(<span class="default"><a href="topalg_1.html#K7" title="TOPALG_1:func.7">pi_1-iso</a> <font color="Maroon" title="c10">P</font></span>)</span> <a href="group_6.html#K7" title="GROUP_6:func.7">*</a> <span class="p1">(<span class="default"><a href="topalg_3.html#K4" title="TOPALG_3:func.4">FundGrIso</a> <font color="Maroon" title="c9">f</font>,<font color="Maroon" title="c6">s</font></span>)</span>
; <span class="comment"><font color="Red">:: according to </font></span><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:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="topalg_1.html#K7" title="TOPALG_1:func.7">pi_1-iso</a> <font color="Maroon" title="c10">P</font></span>)</span> <a href="group_6.html#K7" title="GROUP_6:func.7">*</a> <span class="p1">(<span class="default"><a href="topalg_3.html#K4" title="TOPALG_3:func.4">FundGrIso</a> <font color="Maroon" title="c9">f</font>,<font color="Maroon" title="c6">s</font></span>)</span> is  <a href="funct_2.html#V3" title="FUNCT_2:attr.3">bijective</a> </span><br/>

<a NAME="E5:44"/>
 <a href="topalg_3.html#K4" title="TOPALG_3:func.4">FundGrIso</a> <font color="Maroon" title="c9">f</font>,<font color="Maroon" title="c6">s</font> is  <a href="funct_2.html#V3" title="FUNCT_2:attr.3">bijective</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:44"><span class="lab"><font color="Green" title="E16">A1</font></span></a>, <a class="ref" href="topalg_3.html#T33" target="_self" title="TOPALG_3:th.33">Th33</a></span>;<br/>
<span class="kw">hence </span><a NAME="E6:44"/>
<span class="p1">(<span class="default"><a href="topalg_1.html#K7" title="TOPALG_1:func.7">pi_1-iso</a> <font color="Maroon" title="c10">P</font></span>)</span> <a href="group_6.html#K7" title="GROUP_6:func.7">*</a> <span class="p1">(<span class="default"><a href="topalg_3.html#K4" title="TOPALG_3:func.4">FundGrIso</a> <font color="Maroon" title="c9">f</font>,<font color="Maroon" title="c6">s</font></span>)</span> is  <a href="funct_2.html#V3" title="FUNCT_2:attr.3">bijective</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="group_6.html#T74" title="GROUP_6:th.74">GROUP_6:74</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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