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<span class="kw">let </span><font color="Maroon" title="c1">G</font>, <font color="Maroon" title="c2">H</font> be   <a href="algstr_0.html#V15" title="ALGSTR_0:attr.15">strict</a>  <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="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c1">G</font> is  <a href="struct_0.html#V7" title="STRUCT_0:attr.7">trivial</a>  &amp; <font color="Maroon" title="c2">H</font> is  <a href="struct_0.html#V7" title="STRUCT_0:attr.7">trivial</a>  implies <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c2">H</font> <a href="group_6.html#R2" title="GROUP_6:pred.2">are_isomorphic</a>  )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:100"/><span class="lab"><font color="Green" title="E58">A1</font></span>: 
<font color="Maroon" title="c1">G</font> is  <a href="struct_0.html#V7" title="STRUCT_0:attr.7">trivial</a> 
 <span class="kw">and </span><br/><a NAME="E2:100"/><span class="lab"><font color="Green" title="E59">A2</font></span>: 
<font color="Maroon" title="c2">H</font> is  <a href="struct_0.html#V7" title="STRUCT_0:attr.7">trivial</a> 
 ; <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="c1">G</font>,<font color="Maroon" title="c2">H</font> <a href="group_6.html#R2" title="GROUP_6:pred.2">are_isomorphic</a> </span><br/>

<span class="kw">take </span>
 <a href="group_6.html#K8" title="GROUP_6:func.8">1:</a> <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c2">H</font>
; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="group_6.html#D15" target="_self" 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">  <a href="group_6.html#K8" title="GROUP_6:func.8">1:</a> <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c2">H</font> is  <a href="funct_2.html#V3" title="FUNCT_2:attr.3">bijective</a> </span><br/>

<a NAME="E3:100"/><span class="lab"><font color="Green" title="E60">A3</font></span>: 
<font color="Maroon" title="c2">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_2.html#K6" title="GROUP_2:func.6">(1).</a> <font color="Maroon" title="c2">H</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:100"><span class="lab"><font color="Green" title="E59">A2</font></span></a>, <a class="ref" href="group_6.html#T13" target="_self" title="GROUP_6:th.13">Th13</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c3">h</font> =  <a href="group_6.html#K8" title="GROUP_6:func.8">1:</a> <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c2">H</font>;<br/>
<a NAME="E4:100"/><span class="lab"><font color="Green" title="E61">A4</font></span>: 
<font color="Maroon" title="c1">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_2.html#K6" title="GROUP_2:func.6">(1).</a> <font color="Maroon" title="c1">G</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:100"><span class="lab"><font color="Green" title="E58">A1</font></span></a>, <a class="ref" href="group_6.html#T13" target="_self" title="GROUP_6:th.13">Th13</a></span>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="100_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c4">a</font>, <font color="Maroon" title="c5">b</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">G</font>; <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="group_6.html#K8" title="GROUP_6:func.8">1:</a> <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c2">H</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="group_6.html#K8" title="GROUP_6:func.8">1:</a> <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c2">H</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">b</font> implies <font color="Maroon" title="c4">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">b</font> )</span><br/><span class="kw">assume </span><a NAME="E1:100_1"/>
<span class="p1">(<span class="default"><a href="group_6.html#K8" title="GROUP_6:func.8">1:</a> <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c2">H</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="group_6.html#K8" title="GROUP_6:func.8">1:</a> <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c2">H</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">b</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">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">b</font></span><br/><a NAME="E2:100_1"/>
<font color="Maroon" title="c4">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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_2.html#K6" title="GROUP_2:func.6">(1).</a> <font color="Maroon" title="c1">G</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:100"><span class="lab"><font color="Green" title="E61">A4</font></span></a></span>;<br/><a NAME="E3:100_1"/><span class="kw">then </span>
<font color="Maroon" title="c4">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Maroon" title="c1">G</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="group_2.html#D7" title="GROUP_2:def.7">GROUP_2:def 7</a></span>;<br/><a NAME="E4:100_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E62">A5</font></span>: 
<font color="Maroon" title="c4">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Maroon" title="c1">G</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>;<br/><a NAME="E5:100_1"/>
<font color="Maroon" title="c5">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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_2.html#K6" title="GROUP_2:func.6">(1).</a> <font color="Maroon" title="c1">G</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:100"><span class="lab"><font color="Green" title="E61">A4</font></span></a></span>;<br/><a NAME="E6:100_1"/><span class="kw">then </span>
<font color="Maroon" title="c5">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Maroon" title="c1">G</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="group_2.html#D7" title="GROUP_2:def.7">GROUP_2:def 7</a></span>;<br/><span class="kw">hence </span><a NAME="E7:100_1"/>
<font color="Maroon" title="c4">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">b</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:100_1"><span class="lab"><font color="Green" title="E62">A5</font></span></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</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/></div><span class="kw">end;</span></div>
<a NAME="E6:100"/><span class="kw">then </span><span class="lab"><font color="Green" title="E62">A6</font></span>: 
 <a href="group_6.html#K8" title="GROUP_6:func.8">1:</a> <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c2">H</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="group_6.html#T1" target="_self" title="GROUP_6:th.1">Th1</a></span>;<br/>
<a NAME="E7:100"/>
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="group_2.html#K6" title="GROUP_2:func.6">(1).</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Maroon" title="c1">G</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="group_2.html#D7" title="GROUP_2:def.7">GROUP_2:def 7</a></span>;<br/>
<span class="kw">then </span> <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="group_6.html#K8" title="GROUP_6:func.8">1:</a> <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c2">H</font></span>)</span> = 
<span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="group_6.html#K8" title="GROUP_6:func.8">1:</a> <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c2">H</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p3">(<span class="default"><a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Maroon" title="c1">G</font></span>)</span></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:100"><span class="lab"><font color="Green" title="E61">A4</font></span></a>, <a class="ref" href="funct_2.html#T62" title="FUNCT_2:th.62">FUNCT_2:62</a></span>
<br/>.= 
<span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Maroon" title="c2">H</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="group_6.html#D8" target="_self" title="GROUP_6:def.8">Def8</a></span>
<br/>.= 
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="group_2.html#K6" title="GROUP_2:func.6">(1).</a> <font color="Maroon" title="c2">H</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="group_2.html#D7" title="GROUP_2:def.7">GROUP_2:def 7</a></span>
;<br/>
<a NAME="E9:100"/><span class="kw">then </span>
 <a href="group_6.html#K8" title="GROUP_6:func.8">1:</a> <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c2">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="#E3:100"><span class="lab"><font color="Green" title="E60">A3</font></span></a>, <a class="ref" href="funct_2.html#D3" title="FUNCT_2:def.3">FUNCT_2:def 3</a></span>;<br/>
<span class="kw">hence </span><a NAME="E10:100"/>
 <a href="group_6.html#K8" title="GROUP_6:func.8">1:</a> <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c2">H</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="#E6:100"><span class="lab"><font color="Green" title="E62">A6</font></span></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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