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<b>let </b><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:()"><i><font color="Red">::  thesis: </font></i></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#NK4" title="TOPALG_1:NK.4">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#NK4" title="TOPALG_1:NK.4">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#K4" title="TOPALG_1:func.4">pi_1-iso</a> <font color="Olive" title="b3">P</font> holds <br/><font color="Olive" title="b4">h</font> <a href="group_6.html#NR3" title="GROUP_6:NR.3">is_isomorphism</a> </span><br/><b>let </b><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:()"><i><font color="Red">::  thesis: </font></i></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#NK4" title="TOPALG_1:NK.4">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#NK4" title="TOPALG_1:NK.4">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#K4" title="TOPALG_1:func.4">pi_1-iso</a> <font color="Olive" title="b1">P</font> holds <br/><font color="Olive" title="b2">h</font> <a href="group_6.html#NR3" title="GROUP_6:NR.3">is_isomorphism</a> </span><br/><b>let </b><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:()"><i><font color="Red">::  thesis: </font></i></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#NK4" title="TOPALG_1:NK.4">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#NK4" title="TOPALG_1:NK.4">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#K4" title="TOPALG_1:func.4">pi_1-iso</a> <font color="Maroon" title="c4">P</font> holds <br/><font color="Olive" title="b1">h</font> <a href="group_6.html#NR3" title="GROUP_6:NR.3">is_isomorphism</a>  )</span><br/>







<b>assume </b><a NAME="E1:70"/><i><font color="Green" title="E44">A1</font></i>: 
<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:()"><i><font color="Red">::  thesis: </font></i></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#NK4" title="TOPALG_1:NK.4">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#NK4" title="TOPALG_1:NK.4">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#K4" title="TOPALG_1:func.4">pi_1-iso</a> <font color="Maroon" title="c4">P</font> holds <br/><font color="Olive" title="b1">h</font> <a href="group_6.html#NR3" title="GROUP_6:NR.3">is_isomorphism</a> </span><br/>

<b>let </b><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#NK4" title="TOPALG_1:NK.4">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#NK4" title="TOPALG_1:NK.4">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:()"><i><font color="Red">::  thesis: </font></i></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#K4" title="TOPALG_1:func.4">pi_1-iso</a> <font color="Maroon" title="c4">P</font> implies <font color="Maroon" title="c5">h</font> <a href="group_6.html#NR3" title="GROUP_6:NR.3">is_isomorphism</a>  )</span><br/>

<b>assume </b><a NAME="E2:70"/><i><font color="Green" title="E45">A2</font></i>: 
<font color="Maroon" title="c5">h</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a>  <a href="topalg_1.html#K4" title="TOPALG_1:func.4">pi_1-iso</a> <font color="Maroon" title="c4">P</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c5">h</font> <a href="group_6.html#NR3" title="GROUP_6:NR.3">is_isomorphism</a> </span><br/>

<a NAME="E3:70"/><i><font color="Green" title="E46">A3</font></i>: 
<font color="Maroon" title="c5">h</font> is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>
 
<b>by </b><i><a class="txt" href="topalg_1.html#E1:70"><i><font color="Green" title="E44">A1</font></i></a>, <a class="txt" href="topalg_1.html#E2:70"><i><font color="Green" title="E45">A2</font></i></a>, <a class="ref" href="topalg_1.html#T52" target="_self" title="TOPALG_1:th.52">Th52</a></i>;<br/>
<a NAME="E4:70"/>
<font color="Maroon" title="c5">h</font> is <a href="funct_2.html#V2" title="FUNCT_2:attr.2">onto</a>
 
<b>by </b><i><a class="txt" href="topalg_1.html#E1:70"><i><font color="Green" title="E44">A1</font></i></a>, <a class="txt" href="topalg_1.html#E2:70"><i><font color="Green" title="E45">A2</font></i></a>, <a class="ref" href="topalg_1.html#T53" target="_self" title="TOPALG_1:th.53">Th53</a></i>;<br/>
<a NAME="E5:70"/><b>then </b>
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c5">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="topalg_1.html#NK4" title="TOPALG_1:NK.4">pi_1</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">x0</font></span>)</span>
 
<b>by </b><i><a class="ref" href="funct_2.html#D3" title="FUNCT_2:def.3">FUNCT_2:def 3</a></i>;<br/>
<b>hence </b><a NAME="E6:70"/>
<font color="Maroon" title="c5">h</font> <a href="group_6.html#NR3" title="GROUP_6:NR.3">is_isomorphism</a> 
 <b>by </b><i><a class="txt" href="topalg_1.html#E3:70"><i><font color="Green" title="E46">A3</font></i></a>, <a class="ref" href="group_6.html#T70" title="GROUP_6:th.70">GROUP_6:70</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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