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<span class="kw">let </span><font color="Maroon" title="c1">I0</font>, <font color="Maroon" title="c2">I</font> be   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>   <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a> ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">F0</font> being   <a href="group_7.html#V2" title="GROUP_7:attr.2">Group-like</a>   <a href="group_7.html#V3" title="GROUP_7:attr.3">associative</a>  <a href="group_7.html#NM1" title="GROUP_7:NM.1">multMagma-Family</a> of <font color="Maroon" title="c1">I0</font><br/>  for <font color="Olive" title="b2">F</font> being   <a href="group_7.html#V2" title="GROUP_7:attr.2">Group-like</a>   <a href="group_7.html#V3" title="GROUP_7:attr.3">associative</a>  <a href="group_7.html#NM1" title="GROUP_7:NM.1">multMagma-Family</a> of <font color="Maroon" title="c2">I</font><br/>  for <font color="Olive" title="b3">H</font>, <font color="Olive" title="b4">K</font> being   <a href="group_1.html#NM1" title="GROUP_1:NM.1">Group</a><br/>  for <font color="Olive" title="b5">q</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">I</font><br/>  for <font color="Olive" title="b6">G0</font> being   <a href="group_6.html#NM2" title="GROUP_6:NM.2">Homomorphism</a> of <span class="p1">(<span class="default"><a href="group_7.html#K2" title="GROUP_7:func.2">product</a> <font color="Olive" title="b1">F0</font></span>)</span>,<font color="Olive" title="b3">H</font>  st  not <font color="Olive" title="b5">q</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c1">I0</font> &amp; <font color="Maroon" title="c2">I</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">I0</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Olive" title="b5">q</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> &amp; <font color="Olive" title="b2">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">F0</font> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p1">(<span class="default"><font color="Olive" title="b5">q</font> <a href="funcop_1.html#K17" title="FUNCOP_1:func.17">.--&gt;</a> <font color="Olive" title="b4">K</font></span>)</span> &amp; <font color="Olive" title="b6">G0</font> is  <a href="funct_2.html#V3" title="FUNCT_2:attr.3">bijective</a>  holds <br/> ex <font color="Olive" title="b7">G</font> being   <a href="group_6.html#NM2" title="GROUP_6:NM.2">Homomorphism</a> of <span class="p1">(<span class="default"><a href="group_7.html#K2" title="GROUP_7:func.2">product</a> <font color="Olive" title="b2">F</font></span>)</span>,<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="Olive" title="b3">H</font>,<font color="Olive" title="b4">K</font></span><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">*&gt;</a></span></span>)</span> st <br/>( <font color="Olive" title="b7">G</font> is  <a href="funct_2.html#V3" title="FUNCT_2:attr.3">bijective</a>  &amp; (  for <font color="Olive" title="b8">x0</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><br/>  for <font color="Olive" title="b9">k</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Olive" title="b4">K</font><br/>  for <font color="Olive" title="b10">h</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Olive" title="b3">H</font>  st <font color="Olive" title="b10">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b6">G0</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b8">x0</font> &amp; <font color="Olive" title="b8">x0</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a>  <a href="group_7.html#K2" title="GROUP_7:func.2">product</a> <font color="Olive" title="b1">F0</font> holds <br/><font color="Olive" title="b7">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b8">x0</font> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p2">(<span class="default"><font color="Olive" title="b5">q</font> <a href="funcop_1.html#K17" title="FUNCOP_1:func.17">.--&gt;</a> <font color="Olive" title="b9">k</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="group_7.html#K5" title="GROUP_7:func.5">&lt;*</a><span class="default"><font color="Olive" title="b10">h</font>,<font color="Olive" title="b9">k</font></span><a href="group_7.html#K5" title="GROUP_7:func.5">*&gt;</a></span> ) )</span><br/><span class="kw">let </span><font color="Maroon" title="c3">F0</font> be   <a href="group_7.html#V2" title="GROUP_7:attr.2">Group-like</a>   <a href="group_7.html#V3" title="GROUP_7:attr.3">associative</a>  <a href="group_7.html#NM1" title="GROUP_7:NM.1">multMagma-Family</a> of <font color="Maroon" title="c1">I0</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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_7.html#V2" title="GROUP_7:attr.2">Group-like</a>   <a href="group_7.html#V3" title="GROUP_7:attr.3">associative</a>  <a href="group_7.html#NM1" title="GROUP_7:NM.1">multMagma-Family</a> of <font color="Maroon" title="c2">I</font><br/>  for <font color="Olive" title="b2">H</font>, <font color="Olive" title="b3">K</font> being   <a href="group_1.html#NM1" title="GROUP_1:NM.1">Group</a><br/>  for <font color="Olive" title="b4">q</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">I</font><br/>  for <font color="Olive" title="b5">G0</font> being   <a href="group_6.html#NM2" title="GROUP_6:NM.2">Homomorphism</a> of <span class="p1">(<span class="default"><a href="group_7.html#K2" title="GROUP_7:func.2">product</a> <font color="Maroon" title="c3">F0</font></span>)</span>,<font color="Olive" title="b2">H</font>  st  not <font color="Olive" title="b4">q</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c1">I0</font> &amp; <font color="Maroon" title="c2">I</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">I0</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Olive" title="b4">q</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> &amp; <font color="Olive" title="b1">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">F0</font> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p1">(<span class="default"><font color="Olive" title="b4">q</font> <a href="funcop_1.html#K17" title="FUNCOP_1:func.17">.--&gt;</a> <font color="Olive" title="b3">K</font></span>)</span> &amp; <font color="Olive" title="b5">G0</font> is  <a href="funct_2.html#V3" title="FUNCT_2:attr.3">bijective</a>  holds <br/> ex <font color="Olive" title="b6">G</font> being   <a href="group_6.html#NM2" title="GROUP_6:NM.2">Homomorphism</a> of <span class="p1">(<span class="default"><a href="group_7.html#K2" title="GROUP_7:func.2">product</a> <font color="Olive" title="b1">F</font></span>)</span>,<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="Olive" title="b2">H</font>,<font color="Olive" title="b3">K</font></span><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">*&gt;</a></span></span>)</span> st <br/>( <font color="Olive" title="b6">G</font> is  <a href="funct_2.html#V3" title="FUNCT_2:attr.3">bijective</a>  &amp; (  for <font color="Olive" title="b7">x0</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><br/>  for <font color="Olive" title="b8">k</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Olive" title="b3">K</font><br/>  for <font color="Olive" title="b9">h</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Olive" title="b2">H</font>  st <font color="Olive" title="b9">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b5">G0</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b7">x0</font> &amp; <font color="Olive" title="b7">x0</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a>  <a href="group_7.html#K2" title="GROUP_7:func.2">product</a> <font color="Maroon" title="c3">F0</font> holds <br/><font color="Olive" title="b6">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b7">x0</font> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p2">(<span class="default"><font color="Olive" title="b4">q</font> <a href="funcop_1.html#K17" title="FUNCOP_1:func.17">.--&gt;</a> <font color="Olive" title="b8">k</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="group_7.html#K5" title="GROUP_7:func.5">&lt;*</a><span class="default"><font color="Olive" title="b9">h</font>,<font color="Olive" title="b8">k</font></span><a href="group_7.html#K5" title="GROUP_7:func.5">*&gt;</a></span> ) )</span><br/><span class="kw">let </span><font color="Maroon" title="c4">F</font> be   <a href="group_7.html#V2" title="GROUP_7:attr.2">Group-like</a>   <a href="group_7.html#V3" title="GROUP_7:attr.3">associative</a>  <a href="group_7.html#NM1" title="GROUP_7:NM.1">multMagma-Family</a> of <font color="Maroon" title="c2">I</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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>, <font color="Olive" title="b2">K</font> being   <a href="group_1.html#NM1" title="GROUP_1:NM.1">Group</a><br/>  for <font color="Olive" title="b3">q</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">I</font><br/>  for <font color="Olive" title="b4">G0</font> being   <a href="group_6.html#NM2" title="GROUP_6:NM.2">Homomorphism</a> of <span class="p1">(<span class="default"><a href="group_7.html#K2" title="GROUP_7:func.2">product</a> <font color="Maroon" title="c3">F0</font></span>)</span>,<font color="Olive" title="b1">H</font>  st  not <font color="Olive" title="b3">q</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c1">I0</font> &amp; <font color="Maroon" title="c2">I</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">I0</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Olive" title="b3">q</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> &amp; <font color="Maroon" title="c4">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">F0</font> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p1">(<span class="default"><font color="Olive" title="b3">q</font> <a href="funcop_1.html#K17" title="FUNCOP_1:func.17">.--&gt;</a> <font color="Olive" title="b2">K</font></span>)</span> &amp; <font color="Olive" title="b4">G0</font> is  <a href="funct_2.html#V3" title="FUNCT_2:attr.3">bijective</a>  holds <br/> ex <font color="Olive" title="b5">G</font> being   <a href="group_6.html#NM2" title="GROUP_6:NM.2">Homomorphism</a> of <span class="p1">(<span class="default"><a href="group_7.html#K2" title="GROUP_7:func.2">product</a> <font color="Maroon" title="c4">F</font></span>)</span>,<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="Olive" title="b1">H</font>,<font color="Olive" title="b2">K</font></span><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">*&gt;</a></span></span>)</span> st <br/>( <font color="Olive" title="b5">G</font> is  <a href="funct_2.html#V3" title="FUNCT_2:attr.3">bijective</a>  &amp; (  for <font color="Olive" title="b6">x0</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><br/>  for <font color="Olive" title="b7">k</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Olive" title="b2">K</font><br/>  for <font color="Olive" title="b8">h</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Olive" title="b1">H</font>  st <font color="Olive" title="b8">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">G0</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b6">x0</font> &amp; <font color="Olive" title="b6">x0</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a>  <a href="group_7.html#K2" title="GROUP_7:func.2">product</a> <font color="Maroon" title="c3">F0</font> holds <br/><font color="Olive" title="b5">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b6">x0</font> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p2">(<span class="default"><font color="Olive" title="b3">q</font> <a href="funcop_1.html#K17" title="FUNCOP_1:func.17">.--&gt;</a> <font color="Olive" title="b7">k</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="group_7.html#K5" title="GROUP_7:func.5">&lt;*</a><span class="default"><font color="Olive" title="b8">h</font>,<font color="Olive" title="b7">k</font></span><a href="group_7.html#K5" title="GROUP_7:func.5">*&gt;</a></span> ) )</span><br/><span class="kw">let </span><font color="Maroon" title="c5">H</font>, <font color="Maroon" title="c6">K</font> be   <a href="group_1.html#NM1" title="GROUP_1:NM.1">Group</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">q</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">I</font><br/>  for <font color="Olive" title="b2">G0</font> being   <a href="group_6.html#NM2" title="GROUP_6:NM.2">Homomorphism</a> of <span class="p1">(<span class="default"><a href="group_7.html#K2" title="GROUP_7:func.2">product</a> <font color="Maroon" title="c3">F0</font></span>)</span>,<font color="Maroon" title="c5">H</font>  st  not <font color="Olive" title="b1">q</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c1">I0</font> &amp; <font color="Maroon" title="c2">I</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">I0</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Olive" title="b1">q</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> &amp; <font color="Maroon" title="c4">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">F0</font> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">q</font> <a href="funcop_1.html#K17" title="FUNCOP_1:func.17">.--&gt;</a> <font color="Maroon" title="c6">K</font></span>)</span> &amp; <font color="Olive" title="b2">G0</font> is  <a href="funct_2.html#V3" title="FUNCT_2:attr.3">bijective</a>  holds <br/> ex <font color="Olive" title="b3">G</font> being   <a href="group_6.html#NM2" title="GROUP_6:NM.2">Homomorphism</a> of <span class="p1">(<span class="default"><a href="group_7.html#K2" title="GROUP_7:func.2">product</a> <font color="Maroon" title="c4">F</font></span>)</span>,<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="c5">H</font>,<font color="Maroon" title="c6">K</font></span><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">*&gt;</a></span></span>)</span> st <br/>( <font color="Olive" title="b3">G</font> is  <a href="funct_2.html#V3" title="FUNCT_2:attr.3">bijective</a>  &amp; (  for <font color="Olive" title="b4">x0</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><br/>  for <font color="Olive" title="b5">k</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c6">K</font><br/>  for <font color="Olive" title="b6">h</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c5">H</font>  st <font color="Olive" title="b6">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">G0</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b4">x0</font> &amp; <font color="Olive" title="b4">x0</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a>  <a href="group_7.html#K2" title="GROUP_7:func.2">product</a> <font color="Maroon" title="c3">F0</font> holds <br/><font color="Olive" title="b3">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b4">x0</font> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p2">(<span class="default"><font color="Olive" title="b1">q</font> <a href="funcop_1.html#K17" title="FUNCOP_1:func.17">.--&gt;</a> <font color="Olive" title="b5">k</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="group_7.html#K5" title="GROUP_7:func.5">&lt;*</a><span class="default"><font color="Olive" title="b6">h</font>,<font color="Olive" title="b5">k</font></span><a href="group_7.html#K5" title="GROUP_7:func.5">*&gt;</a></span> ) )</span><br/><span class="kw">let </span><font color="Maroon" title="c7">q</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">I</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">G0</font> being   <a href="group_6.html#NM2" title="GROUP_6:NM.2">Homomorphism</a> of <span class="p1">(<span class="default"><a href="group_7.html#K2" title="GROUP_7:func.2">product</a> <font color="Maroon" title="c3">F0</font></span>)</span>,<font color="Maroon" title="c5">H</font>  st  not <font color="Maroon" title="c7">q</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c1">I0</font> &amp; <font color="Maroon" title="c2">I</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">I0</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c7">q</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> &amp; <font color="Maroon" title="c4">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">F0</font> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">q</font> <a href="funcop_1.html#K17" title="FUNCOP_1:func.17">.--&gt;</a> <font color="Maroon" title="c6">K</font></span>)</span> &amp; <font color="Olive" title="b1">G0</font> is  <a href="funct_2.html#V3" title="FUNCT_2:attr.3">bijective</a>  holds <br/> ex <font color="Olive" title="b2">G</font> being   <a href="group_6.html#NM2" title="GROUP_6:NM.2">Homomorphism</a> of <span class="p1">(<span class="default"><a href="group_7.html#K2" title="GROUP_7:func.2">product</a> <font color="Maroon" title="c4">F</font></span>)</span>,<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="c5">H</font>,<font color="Maroon" title="c6">K</font></span><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">*&gt;</a></span></span>)</span> st <br/>( <font color="Olive" title="b2">G</font> is  <a href="funct_2.html#V3" title="FUNCT_2:attr.3">bijective</a>  &amp; (  for <font color="Olive" title="b3">x0</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><br/>  for <font color="Olive" title="b4">k</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c6">K</font><br/>  for <font color="Olive" title="b5">h</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c5">H</font>  st <font color="Olive" title="b5">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">G0</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b3">x0</font> &amp; <font color="Olive" title="b3">x0</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a>  <a href="group_7.html#K2" title="GROUP_7:func.2">product</a> <font color="Maroon" title="c3">F0</font> holds <br/><font color="Olive" title="b2">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b3">x0</font> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">q</font> <a href="funcop_1.html#K17" title="FUNCOP_1:func.17">.--&gt;</a> <font color="Olive" title="b4">k</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="group_7.html#K5" title="GROUP_7:func.5">&lt;*</a><span class="default"><font color="Olive" title="b5">h</font>,<font color="Olive" title="b4">k</font></span><a href="group_7.html#K5" title="GROUP_7:func.5">*&gt;</a></span> ) )</span><br/><span class="kw">let </span><font color="Maroon" title="c8">G0</font> be   <a href="group_6.html#NM2" title="GROUP_6:NM.2">Homomorphism</a> of <span class="p1">(<span class="default"><a href="group_7.html#K2" title="GROUP_7:func.2">product</a> <font color="Maroon" title="c3">F0</font></span>)</span>,<font color="Maroon" title="c5">H</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c7">q</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c1">I0</font> &amp; <font color="Maroon" title="c2">I</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">I0</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c7">q</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> &amp; <font color="Maroon" title="c4">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">F0</font> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">q</font> <a href="funcop_1.html#K17" title="FUNCOP_1:func.17">.--&gt;</a> <font color="Maroon" title="c6">K</font></span>)</span> &amp; <font color="Maroon" title="c8">G0</font> is  <a href="funct_2.html#V3" title="FUNCT_2:attr.3">bijective</a>  implies  ex <font color="Olive" title="b1">G</font> being   <a href="group_6.html#NM2" title="GROUP_6:NM.2">Homomorphism</a> of <span class="p1">(<span class="default"><a href="group_7.html#K2" title="GROUP_7:func.2">product</a> <font color="Maroon" title="c4">F</font></span>)</span>,<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="c5">H</font>,<font color="Maroon" title="c6">K</font></span><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">*&gt;</a></span></span>)</span> st <br/>( <font color="Olive" title="b1">G</font> is  <a href="funct_2.html#V3" title="FUNCT_2:attr.3">bijective</a>  &amp; (  for <font color="Olive" title="b2">x0</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><br/>  for <font color="Olive" title="b3">k</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c6">K</font><br/>  for <font color="Olive" title="b4">h</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c5">H</font>  st <font color="Olive" title="b4">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">G0</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">x0</font> &amp; <font color="Olive" title="b2">x0</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a>  <a href="group_7.html#K2" title="GROUP_7:func.2">product</a> <font color="Maroon" title="c3">F0</font> holds <br/><font color="Olive" title="b1">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">x0</font> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">q</font> <a href="funcop_1.html#K17" title="FUNCOP_1:func.17">.--&gt;</a> <font color="Olive" title="b3">k</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="group_7.html#K5" title="GROUP_7:func.5">&lt;*</a><span class="default"><font color="Olive" title="b4">h</font>,<font color="Olive" title="b3">k</font></span><a href="group_7.html#K5" title="GROUP_7:func.5">*&gt;</a></span> ) ) )</span><br/>











<span class="kw">assume </span><a NAME="E1:36"/><span class="lab"><font color="Green" title="E37">A1</font></span>: 
(  not <font color="Maroon" title="c7">q</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c1">I0</font> &amp; <font color="Maroon" title="c2">I</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">I0</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c7">q</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> &amp; <font color="Maroon" title="c4">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">F0</font> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">q</font> <a href="funcop_1.html#K17" title="FUNCOP_1:func.17">.--&gt;</a> <font color="Maroon" title="c6">K</font></span>)</span> &amp; <font color="Maroon" title="c8">G0</font> is  <a href="funct_2.html#V3" title="FUNCT_2:attr.3">bijective</a>  )
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">G</font> being   <a href="group_6.html#NM2" title="GROUP_6:NM.2">Homomorphism</a> of <span class="p1">(<span class="default"><a href="group_7.html#K2" title="GROUP_7:func.2">product</a> <font color="Maroon" title="c4">F</font></span>)</span>,<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="c5">H</font>,<font color="Maroon" title="c6">K</font></span><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">*&gt;</a></span></span>)</span> st <br/>( <font color="Olive" title="b1">G</font> is  <a href="funct_2.html#V3" title="FUNCT_2:attr.3">bijective</a>  &amp; (  for <font color="Olive" title="b2">x0</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><br/>  for <font color="Olive" title="b3">k</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c6">K</font><br/>  for <font color="Olive" title="b4">h</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c5">H</font>  st <font color="Olive" title="b4">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">G0</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">x0</font> &amp; <font color="Olive" title="b2">x0</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a>  <a href="group_7.html#K2" title="GROUP_7:func.2">product</a> <font color="Maroon" title="c3">F0</font> holds <br/><font color="Olive" title="b1">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">x0</font> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">q</font> <a href="funcop_1.html#K17" title="FUNCOP_1:func.17">.--&gt;</a> <font color="Olive" title="b3">k</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="group_7.html#K5" title="GROUP_7:func.5">&lt;*</a><span class="default"><font color="Olive" title="b4">h</font>,<font color="Olive" title="b3">k</font></span><a href="group_7.html#K5" title="GROUP_7:func.5">*&gt;</a></span> ) )</span><br/>

<span class="kw">set </span><font color="Maroon" title="c9">L0</font> = <font color="Maroon" title="c8">G0</font> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a> ;<br/>
<a NAME="E2:36"/><span class="lab"><font color="Green" title="E38">A2</font></span>: 
 <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c8">G0</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="c5">H</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D3" onmouseover="rs('funct_2/D3')" onmouseout="rh()">FUNCT_2:def 3</a>, <a class="txt" href="#E1:36"><span class="lab"><font color="Green" title="E37">A1</font></span></a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c10">L0</font> = <font color="Maroon" title="c8">G0</font> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a>  as   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c5">H</font>,<span class="p1">(<span class="default"><a href="group_7.html#K2" title="GROUP_7:func.2">product</a> <font color="Maroon" title="c3">F0</font></span>)</span> <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#T25" onmouseover="rs('funct_2/T25')" onmouseout="rh()">FUNCT_2:25</a>, <a class="txt" href="#E1:36"><span class="lab"><font color="Green" title="E37">A1</font></span></a></span>;<br/>
<a NAME="E4:36"/><span class="lab"><font color="Green" title="E39">A3</font></span>: 
( <font color="Maroon" title="c10">L0</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c8">G0</font> <a href="pboole.html#R8" title="PBOOLE:pred.8">=</a>  <a href="partfun1.html#K6" title="PARTFUN1:func.6">id</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> <font color="Maroon" title="c3">F0</font></span>)</span> &amp; <font color="Maroon" title="c8">G0</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c10">L0</font> <a href="pboole.html#R8" title="PBOOLE:pred.8">=</a>  <a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c5">H</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:36"><span class="lab"><font color="Green" title="E37">A1</font></span></a>, <a class="txt" href="#E2:36"><span class="lab"><font color="Green" title="E38">A2</font></span></a>, <a class="ref" href="funct_2.html#T29" onmouseover="rs('funct_2/T29')" onmouseout="rh()">FUNCT_2:29</a></span>;<br/>
<a NAME="E5:36"/><span class="lab"><font color="Green" title="E40">A4</font></span>: 
<font color="Maroon" title="c10">L0</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="#E4:36"><span class="lab"><font color="Green" title="E39">A3</font></span></a>, <a class="ref" href="funct_2.html#T23" onmouseover="rs('funct_2/T23')" onmouseout="rh()">FUNCT_2:23</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c11">L0</font> = <font color="Maroon" title="c10">L0</font> as   <a href="group_6.html#NM2" title="GROUP_6:NM.2">Homomorphism</a> of <font color="Maroon" title="c5">H</font>,<span class="p1">(<span class="default"><a href="group_7.html#K2" title="GROUP_7:func.2">product</a> <font color="Maroon" title="c3">F0</font></span>)</span> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:36"><span class="lab"><font color="Green" title="E37">A1</font></span></a>, <a class="ref" href="group_6.html#T62" onmouseover="rs('group_6/T62')" onmouseout="rh()">GROUP_6:62</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c12">L</font> being   <a href="group_6.html#NM2" title="GROUP_6:NM.2">Homomorphism</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="c5">H</font>,<font color="Maroon" title="c6">K</font></span><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">*&gt;</a></span></span>)</span>,<span class="p1">(<span class="default"><a href="group_7.html#K2" title="GROUP_7:func.2">product</a> <font color="Maroon" title="c4">F</font></span>)</span><span class="kw"> such that </span><br/><a NAME="E8:36"/><span class="lab"><font color="Green" title="E41">A5</font></span>: 
( <font color="Maroon" title="c12">L</font> is  <a href="funct_2.html#V3" title="FUNCT_2:attr.3">bijective</a>  &amp; (  for <font color="Olive" title="b1">h</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c5">H</font><br/>  for <font color="Olive" title="b2">k</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c6">K</font>  ex <font color="Olive" title="b3">g</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a> st <br/>( <font color="Olive" title="b3">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">L0</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">h</font> &amp; <font color="Maroon" title="c12">L</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1"><a href="group_7.html#K5" title="GROUP_7:func.5">&lt;*</a><span class="default"><font color="Olive" title="b1">h</font>,<font color="Olive" title="b2">k</font></span><a href="group_7.html#K5" title="GROUP_7:func.5">*&gt;</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">g</font> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">q</font> <a href="funcop_1.html#K17" title="FUNCOP_1:func.17">.--&gt;</a> <font color="Olive" title="b2">k</font></span>)</span> ) ) )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="group_17.html#T27" target="_self" onmouseover="rs('group_17/T27')" onmouseout="rh()">Th27</a>, <a class="txt" href="#E1:36"><span class="lab"><font color="Green" title="E37">A1</font></span></a>, <a class="txt" href="#E5:36"><span class="lab"><font color="Green" title="E40">A4</font></span></a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c13">G</font> = <font color="Maroon" title="c12">L</font> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a> ;<br/>
<a NAME="E9:36"/><span class="lab"><font color="Green" title="E42">A6</font></span>: 
 <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c12">L</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="group_7.html#K2" title="GROUP_7:func.2">product</a> <font color="Maroon" title="c4">F</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D3" onmouseover="rs('funct_2/D3')" onmouseout="rh()">FUNCT_2:def 3</a>, <a class="txt" href="#E8:36"><span class="lab"><font color="Green" title="E41">A5</font></span></a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c14">G</font> = <font color="Maroon" title="c12">L</font> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a>  as   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <span class="p1">(<span class="default"><a href="group_7.html#K2" title="GROUP_7:func.2">product</a> <font color="Maroon" title="c4">F</font></span>)</span>,<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="c5">H</font>,<font color="Maroon" title="c6">K</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#T25" onmouseover="rs('funct_2/T25')" onmouseout="rh()">FUNCT_2:25</a>, <a class="txt" href="#E8:36"><span class="lab"><font color="Green" title="E41">A5</font></span></a></span>;<br/>
<a NAME="E11:36"/><span class="lab"><font color="Green" title="E43">A7</font></span>: 
( <font color="Maroon" title="c14">G</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c12">L</font> <a href="pboole.html#R8" title="PBOOLE:pred.8">=</a>  <a href="partfun1.html#K6" title="PARTFUN1:func.6">id</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="c5">H</font>,<font color="Maroon" title="c6">K</font></span><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">*&gt;</a></span></span>)</span> &amp; <font color="Maroon" title="c12">L</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c14">G</font> <a href="pboole.html#R8" title="PBOOLE:pred.8">=</a>  <a href="partfun1.html#K6" title="PARTFUN1:func.6">id</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> <font color="Maroon" title="c4">F</font></span>)</span> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:36"><span class="lab"><font color="Green" title="E41">A5</font></span></a>, <a class="txt" href="#E9:36"><span class="lab"><font color="Green" title="E42">A6</font></span></a>, <a class="ref" href="funct_2.html#T29" onmouseover="rs('funct_2/T29')" onmouseout="rh()">FUNCT_2:29</a></span>;<br/>
<a NAME="E12:36"/><span class="lab"><font color="Green" title="E44">A8</font></span>: 
<font color="Maroon" title="c14">G</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="#E11:36"><span class="lab"><font color="Green" title="E43">A7</font></span></a>, <a class="ref" href="funct_2.html#T23" onmouseover="rs('funct_2/T23')" onmouseout="rh()">FUNCT_2:23</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c15">G</font> = <font color="Maroon" title="c14">G</font> as   <a href="group_6.html#NM2" title="GROUP_6:NM.2">Homomorphism</a> of <span class="p1">(<span class="default"><a href="group_7.html#K2" title="GROUP_7:func.2">product</a> <font color="Maroon" title="c4">F</font></span>)</span>,<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="c5">H</font>,<font color="Maroon" title="c6">K</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="txt" href="#E8:36"><span class="lab"><font color="Green" title="E41">A5</font></span></a>, <a class="ref" href="group_6.html#T62" onmouseover="rs('group_6/T62')" onmouseout="rh()">GROUP_6:62</a></span>;<br/>
<a NAME="E14:36"/>
 for <font color="Olive" title="b1">x0</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><br/>  for <font color="Olive" title="b2">k</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c6">K</font><br/>  for <font color="Olive" title="b3">h</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c5">H</font>  st <font color="Olive" title="b3">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">G0</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x0</font> &amp; <font color="Olive" title="b1">x0</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a>  <a href="group_7.html#K2" title="GROUP_7:func.2">product</a> <font color="Maroon" title="c3">F0</font> holds <br/><font color="Maroon" title="c15">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">x0</font> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">q</font> <a href="funcop_1.html#K17" title="FUNCOP_1:func.17">.--&gt;</a> <font color="Olive" title="b2">k</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="group_7.html#K5" title="GROUP_7:func.5">&lt;*</a><span class="default"><font color="Olive" title="b3">h</font>,<font color="Olive" title="b2">k</font></span><a href="group_7.html#K5" title="GROUP_7:func.5">*&gt;</a></span>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="36_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c16">x0</font> be   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">k</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c6">K</font><br/>  for <font color="Olive" title="b2">h</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c5">H</font>  st <font color="Olive" title="b2">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">G0</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">x0</font> &amp; <font color="Maroon" title="c16">x0</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a>  <a href="group_7.html#K2" title="GROUP_7:func.2">product</a> <font color="Maroon" title="c3">F0</font> holds <br/><font color="Maroon" title="c15">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">x0</font> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">q</font> <a href="funcop_1.html#K17" title="FUNCOP_1:func.17">.--&gt;</a> <font color="Olive" title="b1">k</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="group_7.html#K5" title="GROUP_7:func.5">&lt;*</a><span class="default"><font color="Olive" title="b2">h</font>,<font color="Olive" title="b1">k</font></span><a href="group_7.html#K5" title="GROUP_7:func.5">*&gt;</a></span></span><br/><span class="kw">let </span><font color="Maroon" title="c17">k</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c6">K</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c5">H</font>  st <font color="Olive" title="b1">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">G0</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">x0</font> &amp; <font color="Maroon" title="c16">x0</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a>  <a href="group_7.html#K2" title="GROUP_7:func.2">product</a> <font color="Maroon" title="c3">F0</font> holds <br/><font color="Maroon" title="c15">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">x0</font> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">q</font> <a href="funcop_1.html#K17" title="FUNCOP_1:func.17">.--&gt;</a> <font color="Maroon" title="c17">k</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="group_7.html#K5" title="GROUP_7:func.5">&lt;*</a><span class="default"><font color="Olive" title="b1">h</font>,<font color="Maroon" title="c17">k</font></span><a href="group_7.html#K5" title="GROUP_7:func.5">*&gt;</a></span></span><br/><span class="kw">let </span><font color="Maroon" title="c18">h</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c5">H</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c18">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">G0</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">x0</font> &amp; <font color="Maroon" title="c16">x0</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a>  <a href="group_7.html#K2" title="GROUP_7:func.2">product</a> <font color="Maroon" title="c3">F0</font> implies <font color="Maroon" title="c15">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">x0</font> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">q</font> <a href="funcop_1.html#K17" title="FUNCOP_1:func.17">.--&gt;</a> <font color="Maroon" title="c17">k</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="group_7.html#K5" title="GROUP_7:func.5">&lt;*</a><span class="default"><font color="Maroon" title="c18">h</font>,<font color="Maroon" title="c17">k</font></span><a href="group_7.html#K5" title="GROUP_7:func.5">*&gt;</a></span> )</span><br/>





<span class="kw">assume </span><a NAME="E1:36_1"/><span class="lab"><font color="Green" title="E45">A9</font></span>: 
( <font color="Maroon" title="c18">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">G0</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">x0</font> &amp; <font color="Maroon" title="c16">x0</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a>  <a href="group_7.html#K2" title="GROUP_7:func.2">product</a> <font color="Maroon" title="c3">F0</font> )
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c15">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">x0</font> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">q</font> <a href="funcop_1.html#K17" title="FUNCOP_1:func.17">.--&gt;</a> <font color="Maroon" title="c17">k</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="group_7.html#K5" title="GROUP_7:func.5">&lt;*</a><span class="default"><font color="Maroon" title="c18">h</font>,<font color="Maroon" title="c17">k</font></span><a href="group_7.html#K5" title="GROUP_7:func.5">*&gt;</a></span></span><br/>

<span class="kw">consider </span><font color="Maroon" title="c19">g</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><span class="kw"> such that </span><br/><a NAME="E3:36_1"/><span class="lab"><font color="Green" title="E46">A10</font></span>: 
( <font color="Maroon" title="c19">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">L0</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c18">h</font> &amp; <font color="Maroon" title="c12">L</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1"><a href="group_7.html#K5" title="GROUP_7:func.5">&lt;*</a><span class="default"><font color="Maroon" title="c18">h</font>,<font color="Maroon" title="c17">k</font></span><a href="group_7.html#K5" title="GROUP_7:func.5">*&gt;</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">g</font> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">q</font> <a href="funcop_1.html#K17" title="FUNCOP_1:func.17">.--&gt;</a> <font color="Maroon" title="c17">k</font></span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:36"><span class="lab"><font color="Green" title="E41">A5</font></span></a></span>;<br/>
<a NAME="E4:36_1"/>
<font color="Maroon" title="c19">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">x0</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:36_1"><span class="lab"><font color="Green" title="E46">A10</font></span></a>, <a class="txt" href="#E1:36"><span class="lab"><font color="Green" title="E37">A1</font></span></a>, <a class="txt" href="#E1:36_1"><span class="lab"><font color="Green" title="E45">A9</font></span></a>, <a class="ref" href="funct_2.html#T26" onmouseover="rs('funct_2/T26')" onmouseout="rh()">FUNCT_2:26</a></span>;<br/>
<span class="kw">hence </span><a NAME="E5:36_1"/>
<font color="Maroon" title="c15">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">x0</font> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">q</font> <a href="funcop_1.html#K17" title="FUNCOP_1:func.17">.--&gt;</a> <font color="Maroon" title="c17">k</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="group_7.html#K5" title="GROUP_7:func.5">&lt;*</a><span class="default"><font color="Maroon" title="c18">h</font>,<font color="Maroon" title="c17">k</font></span><a href="group_7.html#K5" title="GROUP_7:func.5">*&gt;</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:36"><span class="lab"><font color="Green" title="E41">A5</font></span></a>, <a class="ref" href="funct_2.html#T26" onmouseover="rs('funct_2/T26')" onmouseout="rh()">FUNCT_2:26</a>, <a class="txt" href="#E3:36_1"><span class="lab"><font color="Green" title="E46">A10</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E15:36"/>
 ex <font color="Olive" title="b1">G</font> being   <a href="group_6.html#NM2" title="GROUP_6:NM.2">Homomorphism</a> of <span class="p1">(<span class="default"><a href="group_7.html#K2" title="GROUP_7:func.2">product</a> <font color="Maroon" title="c4">F</font></span>)</span>,<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="c5">H</font>,<font color="Maroon" title="c6">K</font></span><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">*&gt;</a></span></span>)</span> st <br/>( <font color="Olive" title="b1">G</font> is  <a href="funct_2.html#V3" title="FUNCT_2:attr.3">bijective</a>  &amp; (  for <font color="Olive" title="b2">x0</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><br/>  for <font color="Olive" title="b3">k</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c6">K</font><br/>  for <font color="Olive" title="b4">h</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c5">H</font>  st <font color="Olive" title="b4">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">G0</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">x0</font> &amp; <font color="Olive" title="b2">x0</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a>  <a href="group_7.html#K2" title="GROUP_7:func.2">product</a> <font color="Maroon" title="c3">F0</font> holds <br/><font color="Olive" title="b1">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">x0</font> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">q</font> <a href="funcop_1.html#K17" title="FUNCOP_1:func.17">.--&gt;</a> <font color="Olive" title="b3">k</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="group_7.html#K5" title="GROUP_7:func.5">&lt;*</a><span class="default"><font color="Olive" title="b4">h</font>,<font color="Olive" title="b3">k</font></span><a href="group_7.html#K5" title="GROUP_7:func.5">*&gt;</a></span> ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:36"><span class="lab"><font color="Green" title="E44">A8</font></span></a>, <a class="txt" href="#E8:36"><span class="lab"><font color="Green" title="E41">A5</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
