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<span class="kw">let </span><font color="Maroon" title="c1">X</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="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">x0</font>, <font color="Olive" title="b2">x1</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">X</font><br/>  for <font color="Olive" title="b3">P</font> being    <a href="borsuk_2.html#M1" title="BORSUK_2:mode.1">Path</a> of <font color="Olive" title="b1">x0</font>,<font color="Olive" title="b2">x1</font>  st <font color="Olive" title="b1">x0</font>,<font color="Olive" title="b2">x1</font> <a href="borsuk_6.html#R1" title="BORSUK_6:pred.1">are_connected</a>  holds <br/> for <font color="Olive" title="b4">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#NK6" title="TOPALG_1:NK.6">pi_1</a> (<font color="Maroon" title="c1">X</font>,<font color="Olive" title="b2">x1</font>)</span>)</span>,<span class="p1">(<span class="default"><a href="topalg_1.html#NK6" title="TOPALG_1:NK.6">pi_1</a> (<font color="Maroon" title="c1">X</font>,<font color="Olive" title="b1">x0</font>)</span>)</span>  st <font color="Olive" title="b4">h</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a>  <a href="topalg_1.html#K6" title="TOPALG_1:func.6">pi_1-iso</a> <font color="Olive" title="b3">P</font> holds <br/><font color="Olive" title="b4">h</font> is  <a href="funct_2.html#V3" title="FUNCT_2:attr.3">bijective</a> </span><br/><span class="kw">let </span><font color="Maroon" title="c2">x0</font>, <font color="Maroon" title="c3">x1</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">X</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">P</font> being    <a href="borsuk_2.html#M1" title="BORSUK_2:mode.1">Path</a> of <font color="Maroon" title="c2">x0</font>,<font color="Maroon" title="c3">x1</font>  st <font color="Maroon" title="c2">x0</font>,<font color="Maroon" title="c3">x1</font> <a href="borsuk_6.html#R1" title="BORSUK_6:pred.1">are_connected</a>  holds <br/> for <font color="Olive" title="b2">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#NK6" title="TOPALG_1:NK.6">pi_1</a> (<font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c3">x1</font>)</span>)</span>,<span class="p1">(<span class="default"><a href="topalg_1.html#NK6" title="TOPALG_1:NK.6">pi_1</a> (<font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">x0</font>)</span>)</span>  st <font color="Olive" title="b2">h</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a>  <a href="topalg_1.html#K6" title="TOPALG_1:func.6">pi_1-iso</a> <font color="Olive" title="b1">P</font> holds <br/><font color="Olive" title="b2">h</font> is  <a href="funct_2.html#V3" title="FUNCT_2:attr.3">bijective</a> </span><br/><span class="kw">let </span><font color="Maroon" title="c4">P</font> be    <a href="borsuk_2.html#M1" title="BORSUK_2:mode.1">Path</a> of <font color="Maroon" title="c2">x0</font>,<font color="Maroon" title="c3">x1</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c2">x0</font>,<font color="Maroon" title="c3">x1</font> <a href="borsuk_6.html#R1" title="BORSUK_6:pred.1">are_connected</a>  implies  for <font color="Olive" title="b1">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#NK6" title="TOPALG_1:NK.6">pi_1</a> (<font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c3">x1</font>)</span>)</span>,<span class="p1">(<span class="default"><a href="topalg_1.html#NK6" title="TOPALG_1:NK.6">pi_1</a> (<font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">x0</font>)</span>)</span>  st <font color="Olive" title="b1">h</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a>  <a href="topalg_1.html#K6" title="TOPALG_1:func.6">pi_1-iso</a> <font color="Maroon" title="c4">P</font> holds <br/><font color="Olive" title="b1">h</font> is  <a href="funct_2.html#V3" title="FUNCT_2:attr.3">bijective</a>  )</span><br/>







<span class="kw">assume </span><a NAME="E1:73"/><span class="lab"><font color="Green" title="E44">A1</font></span>: 
<font color="Maroon" title="c2">x0</font>,<font color="Maroon" title="c3">x1</font> <a href="borsuk_6.html#R1" title="BORSUK_6:pred.1">are_connected</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">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#NK6" title="TOPALG_1:NK.6">pi_1</a> (<font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c3">x1</font>)</span>)</span>,<span class="p1">(<span class="default"><a href="topalg_1.html#NK6" title="TOPALG_1:NK.6">pi_1</a> (<font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">x0</font>)</span>)</span>  st <font color="Olive" title="b1">h</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a>  <a href="topalg_1.html#K6" title="TOPALG_1:func.6">pi_1-iso</a> <font color="Maroon" title="c4">P</font> holds <br/><font color="Olive" title="b1">h</font> is  <a href="funct_2.html#V3" title="FUNCT_2:attr.3">bijective</a> </span><br/>

<span class="kw">let </span><font color="Maroon" title="c5">h</font> be   <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#NK6" title="TOPALG_1:NK.6">pi_1</a> (<font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c3">x1</font>)</span>)</span>,<span class="p1">(<span class="default"><a href="topalg_1.html#NK6" title="TOPALG_1:NK.6">pi_1</a> (<font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">x0</font>)</span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c5">h</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a>  <a href="topalg_1.html#K6" title="TOPALG_1:func.6">pi_1-iso</a> <font color="Maroon" title="c4">P</font> implies <font color="Maroon" title="c5">h</font> is  <a href="funct_2.html#V3" title="FUNCT_2:attr.3">bijective</a>  )</span><br/>

<span class="kw">assume </span><a NAME="E2:73"/>
<font color="Maroon" title="c5">h</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a>  <a href="topalg_1.html#K6" title="TOPALG_1:func.6">pi_1-iso</a> <font color="Maroon" title="c4">P</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c5">h</font> is  <a href="funct_2.html#V3" title="FUNCT_2:attr.3">bijective</a> </span><br/>

<a NAME="E3:73"/><span class="kw">then </span>
( <font color="Maroon" title="c5">h</font> is  <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>  &amp; <font color="Maroon" title="c5">h</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="#E1:73"><span class="lab"><font color="Green" title="E44">A1</font></span></a>, <a class="ref" href="topalg_1.html#T51" target="_self" title="TOPALG_1:th.51">Th52</a>, <a class="ref" href="topalg_1.html#T52" target="_self" title="TOPALG_1:th.52">Th53</a></span>;<br/>
<span class="kw">hence </span><a NAME="E4:73"/>
<font color="Maroon" title="c5">h</font> is  <a href="funct_2.html#V3" title="FUNCT_2:attr.3">bijective</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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