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<span class="kw">let </span><font color="Maroon" title="c1">G1</font>, <font color="Maroon" title="c2">G2</font>, <font color="Maroon" title="c3">H1</font>, <font color="Maroon" title="c4">H2</font> be   <a href="group_1.html#NM1" title="GROUP_1:NM.1">Group</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">f</font> being   <a href="group_6.html#NM2" title="GROUP_6:NM.2">Homomorphism</a> of <font color="Maroon" title="c1">G1</font>,<font color="Maroon" title="c3">H1</font><br/>  for <font color="Olive" title="b2">g</font> being   <a href="group_6.html#NM2" title="GROUP_6:NM.2">Homomorphism</a> of <font color="Maroon" title="c2">G2</font>,<font color="Maroon" title="c4">H2</font>  st <font color="Olive" title="b1">f</font> is  <a href="funct_2.html#V3" title="FUNCT_2:attr.3">bijective</a>  &amp; <font color="Olive" title="b2">g</font> is  <a href="funct_2.html#V3" title="FUNCT_2:attr.3">bijective</a>  holds <br/> <a href="topalg_4.html#K1" title="TOPALG_4:func.1">Gr2Iso</a> (<font color="Olive" title="b1">f</font>,<font color="Olive" title="b2">g</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">h1</font> be   <a href="group_6.html#NM2" title="GROUP_6:NM.2">Homomorphism</a> of <font color="Maroon" title="c1">G1</font>,<font color="Maroon" title="c3">H1</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">g</font> being   <a href="group_6.html#NM2" title="GROUP_6:NM.2">Homomorphism</a> of <font color="Maroon" title="c2">G2</font>,<font color="Maroon" title="c4">H2</font>  st <font color="Maroon" title="c5">h1</font> is  <a href="funct_2.html#V3" title="FUNCT_2:attr.3">bijective</a>  &amp; <font color="Olive" title="b1">g</font> is  <a href="funct_2.html#V3" title="FUNCT_2:attr.3">bijective</a>  holds <br/> <a href="topalg_4.html#K1" title="TOPALG_4:func.1">Gr2Iso</a> (<font color="Maroon" title="c5">h1</font>,<font color="Olive" title="b1">g</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="c6">h2</font> be   <a href="group_6.html#NM2" title="GROUP_6:NM.2">Homomorphism</a> of <font color="Maroon" title="c2">G2</font>,<font color="Maroon" title="c4">H2</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">h1</font> is  <a href="funct_2.html#V3" title="FUNCT_2:attr.3">bijective</a>  &amp; <font color="Maroon" title="c6">h2</font> is  <a href="funct_2.html#V3" title="FUNCT_2:attr.3">bijective</a>  implies  <a href="topalg_4.html#K1" title="TOPALG_4:func.1">Gr2Iso</a> (<font color="Maroon" title="c5">h1</font>,<font color="Maroon" title="c6">h2</font>) is  <a href="funct_2.html#V3" title="FUNCT_2:attr.3">bijective</a>  )</span><br/>





<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:7"/><span class="lab"><font color="Green" title="E8">A1</font></span>: 
<font color="Maroon" title="c5">h1</font> is  <a href="funct_2.html#V3" title="FUNCT_2:attr.3">bijective</a> 
 <span class="kw">and </span><br/><a NAME="E2:7"/><span class="lab"><font color="Green" title="E9">A2</font></span>: 
<font color="Maroon" title="c6">h2</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">  <a href="topalg_4.html#K1" title="TOPALG_4:func.1">Gr2Iso</a> (<font color="Maroon" title="c5">h1</font>,<font color="Maroon" title="c6">h2</font>) is  <a href="funct_2.html#V3" title="FUNCT_2:attr.3">bijective</a> </span><br/>

<span class="kw">set </span><font color="Maroon" title="c7">h</font> =  <a href="topalg_4.html#K1" title="TOPALG_4:func.1">Gr2Iso</a> (<font color="Maroon" title="c5">h1</font>,<font color="Maroon" title="c6">h2</font>);<br/>
<a NAME="E3:7"/>
 <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c6">h2</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 <font color="Maroon" title="c4">H2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:7"><span class="lab"><font color="Green" title="E9">A2</font></span></a>, <a class="ref" href="group_6.html#T60" title="GROUP_6:th.60">GROUP_6:60</a></span>;<br/>
<a NAME="E4:7"/><span class="kw">then </span><span class="lab"><font color="Green" title="E10">A3</font></span>: 
<font color="Maroon" title="c6">h2</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="ref" href="funct_2.html#D3" title="FUNCT_2:def.3">FUNCT_2:def 3</a></span>;<br/>
<a NAME="E5:7"/>
 <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c5">h1</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 <font color="Maroon" title="c3">H1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:7"><span class="lab"><font color="Green" title="E8">A1</font></span></a>, <a class="ref" href="group_6.html#T60" title="GROUP_6:th.60">GROUP_6:60</a></span>;<br/>
<a NAME="E6:7"/><span class="kw">then </span>
<font color="Maroon" title="c5">h1</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="ref" href="funct_2.html#D3" title="FUNCT_2:def.3">FUNCT_2:def 3</a></span>;<br/>
<a NAME="E7:7"/><span class="kw">then </span>
 <a href="topalg_4.html#K1" title="TOPALG_4:func.1">Gr2Iso</a> (<font color="Maroon" title="c5">h1</font>,<font color="Maroon" title="c6">h2</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="#E4:7"><span class="lab"><font color="Green" title="E10">A3</font></span></a>, <a class="ref" href="topalg_4.html#T4" target="_self" title="TOPALG_4:th.4">Th4</a></span>;<br/>
<a NAME="E8:7"/><span class="kw">then </span><span class="lab"><font color="Green" title="E11">A4</font></span>: 
 <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="topalg_4.html#K1" title="TOPALG_4:func.1">Gr2Iso</a> (<font color="Maroon" title="c5">h1</font>,<font color="Maroon" title="c6">h2</font>)</span>)</span> <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="group_7.html#K2" title="GROUP_7:func.2">product</a> <span class="p2"><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">&lt;*</a><span class="default"><font color="Maroon" title="c3">H1</font>,<font color="Maroon" title="c4">H2</font></span><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">*&gt;</a></span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D3" title="FUNCT_2:def.3">FUNCT_2:def 3</a></span>;<br/>
<a NAME="E9:7"/>
( <font color="Maroon" title="c5">h1</font> is  <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>  &amp; <font color="Maroon" title="c6">h2</font> is  <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>  )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:7"><span class="lab"><font color="Green" title="E8">A1</font></span></a>, <a class="txt" href="#E2:7"><span class="lab"><font color="Green" title="E9">A2</font></span></a>, <a class="ref" href="group_6.html#T60" title="GROUP_6:th.60">GROUP_6:60</a></span>;<br/>
<a NAME="E10:7"/><span class="kw">then </span>
 <a href="topalg_4.html#K1" title="TOPALG_4:func.1">Gr2Iso</a> (<font color="Maroon" title="c5">h1</font>,<font color="Maroon" title="c6">h2</font>) is  <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="topalg_4.html#T3" target="_self" title="TOPALG_4:th.3">Th3</a></span>;<br/>
<span class="kw">hence </span><a NAME="E11:7"/>
 <a href="topalg_4.html#K1" title="TOPALG_4:func.1">Gr2Iso</a> (<font color="Maroon" title="c5">h1</font>,<font color="Maroon" title="c6">h2</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="#E8:7"><span class="lab"><font color="Green" title="E11">A4</font></span></a>, <a class="ref" href="group_6.html#T60" title="GROUP_6:th.60">GROUP_6:60</a></span>; <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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