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<span class="kw">let </span><font color="Maroon" title="c1">G</font>, <font color="Maroon" title="c2">H</font>, <font color="Maroon" title="c3">I</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">h</font> being   <a href="group_6.html#NM2" title="GROUP_6:NM.2">Homomorphism</a> of <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c2">H</font><br/>  for <font color="Olive" title="b2">h1</font> being   <a href="group_6.html#NM2" title="GROUP_6:NM.2">Homomorphism</a> of <font color="Maroon" title="c2">H</font>,<font color="Maroon" title="c3">I</font> holds <br/> ( <font color="Olive" title="b2">h1</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p1">(<span class="default"><a href="group_6.html#K7" title="GROUP_6:func.7">1:</a> (<font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c2">H</font>)</span>)</span> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a>  <a href="group_6.html#K7" title="GROUP_6:func.7">1:</a> (<font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c3">I</font>) &amp; <span class="p1">(<span class="default"><a href="group_6.html#K7" title="GROUP_6:func.7">1:</a> (<font color="Maroon" title="c2">H</font>,<font color="Maroon" title="c3">I</font>)</span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Olive" title="b1">h</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a>  <a href="group_6.html#K7" title="GROUP_6:func.7">1:</a> (<font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c3">I</font>) )</span><br/><span class="kw">let </span><font color="Maroon" title="c4">h</font> be   <a href="group_6.html#NM2" title="GROUP_6:NM.2">Homomorphism</a> of <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c2">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">  for <font color="Olive" title="b1">h1</font> being   <a href="group_6.html#NM2" title="GROUP_6:NM.2">Homomorphism</a> of <font color="Maroon" title="c2">H</font>,<font color="Maroon" title="c3">I</font> holds <br/> ( <font color="Olive" title="b1">h1</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p1">(<span class="default"><a href="group_6.html#K7" title="GROUP_6:func.7">1:</a> (<font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c2">H</font>)</span>)</span> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a>  <a href="group_6.html#K7" title="GROUP_6:func.7">1:</a> (<font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c3">I</font>) &amp; <span class="p1">(<span class="default"><a href="group_6.html#K7" title="GROUP_6:func.7">1:</a> (<font color="Maroon" title="c2">H</font>,<font color="Maroon" title="c3">I</font>)</span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c4">h</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a>  <a href="group_6.html#K7" title="GROUP_6:func.7">1:</a> (<font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c3">I</font>) )</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="c2">H</font>,<font color="Maroon" title="c3">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"> ( <font color="Maroon" title="c5">h1</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p1">(<span class="default"><a href="group_6.html#K7" title="GROUP_6:func.7">1:</a> (<font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c2">H</font>)</span>)</span> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a>  <a href="group_6.html#K7" title="GROUP_6:func.7">1:</a> (<font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c3">I</font>) &amp; <span class="p1">(<span class="default"><a href="group_6.html#K7" title="GROUP_6:func.7">1:</a> (<font color="Maroon" title="c2">H</font>,<font color="Maroon" title="c3">I</font>)</span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c4">h</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a>  <a href="group_6.html#K7" title="GROUP_6:func.7">1:</a> (<font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c3">I</font>) )</span><br/>









<span class="kw">thus </span><a NAME="E1:57"/>
<font color="Maroon" title="c5">h1</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p1">(<span class="default"><a href="group_6.html#K7" title="GROUP_6:func.7">1:</a> (<font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c2">H</font>)</span>)</span> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a>  <a href="group_6.html#K7" title="GROUP_6:func.7">1:</a> (<font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c3">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"> <span class="p1">(<span class="default"><a href="group_6.html#K7" title="GROUP_6:func.7">1:</a> (<font color="Maroon" title="c2">H</font>,<font color="Maroon" title="c3">I</font>)</span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c4">h</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a>  <a href="group_6.html#K7" title="GROUP_6:func.7">1:</a> (<font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c3">I</font>)</span><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="57_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c6">a</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">G</font>; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="funct_2.html#D8" onmouseover="rs('funct_2/D8')" onmouseout="rh()">FUNCT_2:def 8</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"> <span class="p1">(<span class="default"><font color="Maroon" title="c5">h1</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><a href="group_6.html#K7" title="GROUP_6:func.7">1:</a> (<font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c2">H</font>)</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="group_6.html#K7" title="GROUP_6:func.7">1:</a> (<font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c3">I</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">a</font></span><br/>

<span class="kw">thus </span><span class="p1">(<span class="default"><font color="Maroon" title="c5">h1</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><a href="group_6.html#K7" title="GROUP_6:func.7">1:</a> (<font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c2">H</font>)</span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">a</font> = 
<font color="Maroon" title="c5">h1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="group_6.html#K7" title="GROUP_6:func.7">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="c6">a</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#T15" onmouseover="rs('funct_2/T15')" onmouseout="rh()">FUNCT_2:15</a></span>
<br/>.= 
 <a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Maroon" title="c3">I</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="group_6.html#T31" target="_self" onmouseover="rs('group_6/T31')" onmouseout="rh()">Th31</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="group_6.html#K7" title="GROUP_6:func.7">1:</a> (<font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c3">I</font>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">a</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"> verum</span><br/>


</div><span class="kw">end;</span></div>

<span class="kw">let </span><font color="Maroon" title="c6">a</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">G</font>; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="funct_2.html#D8" onmouseover="rs('funct_2/D8')" onmouseout="rh()">FUNCT_2:def 8</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"> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="group_6.html#K7" title="GROUP_6:func.7">1:</a> (<font color="Maroon" title="c2">H</font>,<font color="Maroon" title="c3">I</font>)</span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c4">h</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="group_6.html#K7" title="GROUP_6:func.7">1:</a> (<font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c3">I</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">a</font></span><br/>

<span class="kw">thus </span><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="group_6.html#K7" title="GROUP_6:func.7">1:</a> (<font color="Maroon" title="c2">H</font>,<font color="Maroon" title="c3">I</font>)</span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c4">h</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">a</font> = 
<span class="p1">(<span class="default"><a href="group_6.html#K7" title="GROUP_6:func.7">1:</a> (<font color="Maroon" title="c2">H</font>,<font color="Maroon" title="c3">I</font>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">h</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">a</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#T15" onmouseover="rs('funct_2/T15')" onmouseout="rh()">FUNCT_2:15</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="group_6.html#K7" title="GROUP_6:func.7">1:</a> (<font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c3">I</font>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">a</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"> verum</span><br/>


</div>
