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<b>let </b><font color="Maroon" title="c3">S1</font>, <font color="Maroon" title="c4">S2</font> be  <a href="grcat_1.html#V2" title="GRCAT_1:attr.2">strict</a> <a href="grcat_1.html#NM1" title="GRCAT_1:NM.1">GroupMorphism</a>;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:42_1_2"/><i><font color="Green" title="E22">A8</font></i>: 
for <font color="Olive" title="b1">G1</font>, <font color="Olive" title="b2">G2</font>, <font color="Olive" title="b3">G3</font> being  <a href="algstr_1.html#NM2" title="ALGSTR_1:NM.2">AddGroup</a><br/> for <font color="Olive" title="b4">g</font> being  <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Olive" title="b2">G2</font>,<font color="Olive" title="b3">G3</font><br/> for <font color="Olive" title="b5">f</font> being  <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Olive" title="b1">G1</font>,<font color="Olive" title="b2">G2</font>  st  <a href="grcat_1.html#G1" title="GRCAT_1:aggr.1">GroupMorphismStr</a>(# the <a href="grcat_1.html#U1" title="GRCAT_1:sel.1">Source</a> of <font color="Maroon" title="c1">G</font>,the <a href="grcat_1.html#U2" title="GRCAT_1:sel.2">Target</a> of <font color="Maroon" title="c1">G</font>,the <a href="grcat_1.html#U3" title="GRCAT_1:sel.3">Fun</a> of <font color="Maroon" title="c1">G</font> #) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="grcat_1.html#G1" title="GRCAT_1:aggr.1">GroupMorphismStr</a>(# <font color="Olive" title="b2">G2</font>,<font color="Olive" title="b3">G3</font>,<font color="Olive" title="b4">g</font> #) &amp;  <a href="grcat_1.html#G1" title="GRCAT_1:aggr.1">GroupMorphismStr</a>(# the <a href="grcat_1.html#U1" title="GRCAT_1:sel.1">Source</a> of <font color="Maroon" title="c2">F</font>,the <a href="grcat_1.html#U2" title="GRCAT_1:sel.2">Target</a> of <font color="Maroon" title="c2">F</font>,the <a href="grcat_1.html#U3" title="GRCAT_1:sel.3">Fun</a> of <font color="Maroon" title="c2">F</font> #) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="grcat_1.html#G1" title="GRCAT_1:aggr.1">GroupMorphismStr</a>(# <font color="Olive" title="b1">G1</font>,<font color="Olive" title="b2">G2</font>,<font color="Olive" title="b5">f</font> #) holds <br/><font color="Maroon" title="c3">S1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="grcat_1.html#G1" title="GRCAT_1:aggr.1">GroupMorphismStr</a>(# <font color="Olive" title="b1">G1</font>,<font color="Olive" title="b3">G3</font>,<span class="p1">(<span class="default"><font color="Olive" title="b4">g</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Olive" title="b5">f</font></span>)</span> #)
 <b>and </b><br/><a NAME="E2:42_1_2"/><i><font color="Green" title="E23">A9</font></i>: 
for <font color="Olive" title="b1">G1</font>, <font color="Olive" title="b2">G2</font>, <font color="Olive" title="b3">G3</font> being  <a href="algstr_1.html#NM2" title="ALGSTR_1:NM.2">AddGroup</a><br/> for <font color="Olive" title="b4">g</font> being  <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Olive" title="b2">G2</font>,<font color="Olive" title="b3">G3</font><br/> for <font color="Olive" title="b5">f</font> being  <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Olive" title="b1">G1</font>,<font color="Olive" title="b2">G2</font>  st  <a href="grcat_1.html#G1" title="GRCAT_1:aggr.1">GroupMorphismStr</a>(# the <a href="grcat_1.html#U1" title="GRCAT_1:sel.1">Source</a> of <font color="Maroon" title="c1">G</font>,the <a href="grcat_1.html#U2" title="GRCAT_1:sel.2">Target</a> of <font color="Maroon" title="c1">G</font>,the <a href="grcat_1.html#U3" title="GRCAT_1:sel.3">Fun</a> of <font color="Maroon" title="c1">G</font> #) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="grcat_1.html#G1" title="GRCAT_1:aggr.1">GroupMorphismStr</a>(# <font color="Olive" title="b2">G2</font>,<font color="Olive" title="b3">G3</font>,<font color="Olive" title="b4">g</font> #) &amp;  <a href="grcat_1.html#G1" title="GRCAT_1:aggr.1">GroupMorphismStr</a>(# the <a href="grcat_1.html#U1" title="GRCAT_1:sel.1">Source</a> of <font color="Maroon" title="c2">F</font>,the <a href="grcat_1.html#U2" title="GRCAT_1:sel.2">Target</a> of <font color="Maroon" title="c2">F</font>,the <a href="grcat_1.html#U3" title="GRCAT_1:sel.3">Fun</a> of <font color="Maroon" title="c2">F</font> #) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="grcat_1.html#G1" title="GRCAT_1:aggr.1">GroupMorphismStr</a>(# <font color="Olive" title="b1">G1</font>,<font color="Olive" title="b2">G2</font>,<font color="Olive" title="b5">f</font> #) holds <br/><font color="Maroon" title="c4">S2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="grcat_1.html#G1" title="GRCAT_1:aggr.1">GroupMorphismStr</a>(# <font color="Olive" title="b1">G1</font>,<font color="Olive" title="b3">G3</font>,<span class="p1">(<span class="default"><font color="Olive" title="b4">g</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Olive" title="b5">f</font></span>)</span> #)
 ;<br/>

<b>consider </b><font color="Maroon" title="c5">G2'</font>, <font color="Maroon" title="c6">G3'</font> being   <a href="algstr_1.html#NM2" title="ALGSTR_1:NM.2">AddGroup</a><b> such that </b><br/><a NAME="E4:42_1_2"/><i><font color="Green" title="E24">A10</font></i>: 
<font color="Maroon" title="c1">G</font> is    <a href="grcat_1.html#M1" title="GRCAT_1:mode.1">Morphism</a> of <font color="Maroon" title="c5">G2'</font>,<font color="Maroon" title="c6">G3'</font>
 <b>by </b><i><a class="ref" href="grcat_1.html#T24" target="_self" title="GRCAT_1:th.24">Th24</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c7">G'</font> = <font color="Maroon" title="c1">G</font> as    <a href="grcat_1.html#M1" title="GRCAT_1:mode.1">Morphism</a> of <font color="Maroon" title="c5">G2'</font>,<font color="Maroon" title="c6">G3'</font> <b>by </b><i><a class="txt" href="grcat_1.html#E4:42_1_2"><i><font color="Green" title="E24">A10</font></i></a></i>;<br/>
<a NAME="E6:42_1_2"/><i><font color="Green" title="E25">A11</font></i>: 
<font color="Maroon" title="c5">G2'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="grcat_1.html#K8" title="GRCAT_1:func.8">dom</a> <font color="Maroon" title="c1">G</font>
 
<b>by </b><i><a class="txt" href="grcat_1.html#E4:42_1_2"><i><font color="Green" title="E24">A10</font></i></a>, <a class="ref" href="grcat_1.html#D19" target="_self" title="GRCAT_1:def.19">Def19</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c8">G1'</font>, <font color="Maroon" title="c9">G2''</font> being   <a href="algstr_1.html#NM2" title="ALGSTR_1:NM.2">AddGroup</a><b> such that </b><br/><a NAME="E8:42_1_2"/><i><font color="Green" title="E26">A12</font></i>: 
<font color="Maroon" title="c2">F</font> is    <a href="grcat_1.html#M1" title="GRCAT_1:mode.1">Morphism</a> of <font color="Maroon" title="c8">G1'</font>,<font color="Maroon" title="c9">G2''</font>
 <b>by </b><i><a class="ref" href="grcat_1.html#T24" target="_self" title="GRCAT_1:th.24">Th24</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c10">F'</font> = <font color="Maroon" title="c2">F</font> as    <a href="grcat_1.html#M1" title="GRCAT_1:mode.1">Morphism</a> of <font color="Maroon" title="c8">G1'</font>,<font color="Maroon" title="c9">G2''</font> <b>by </b><i><a class="txt" href="grcat_1.html#E8:42_1_2"><i><font color="Green" title="E26">A12</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c11">F'</font> = <font color="Maroon" title="c10">F'</font> as    <a href="grcat_1.html#M1" title="GRCAT_1:mode.1">Morphism</a> of <font color="Maroon" title="c8">G1'</font>,<font color="Maroon" title="c5">G2'</font> <b>by </b><i><a class="txt" href="grcat_1.html#E1:42"><i><font color="Green" title="E21">A1</font></i></a>, <a class="txt" href="grcat_1.html#E6:42_1_2"><i><font color="Green" title="E25">A11</font></i></a>, <a class="ref" href="grcat_1.html#D19" target="_self" title="GRCAT_1:def.19">Def19</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c12">g'</font> being   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c5">G2'</font>,<font color="Maroon" title="c6">G3'</font><b> such that </b><br/><a NAME="E12:42_1_2"/><i><font color="Green" title="E27">A13</font></i>: 
 <a href="grcat_1.html#G1" title="GRCAT_1:aggr.1">GroupMorphismStr</a>(# the <a href="grcat_1.html#U1" title="GRCAT_1:sel.1">Source</a> of <font color="Maroon" title="c7">G'</font>,the <a href="grcat_1.html#U2" title="GRCAT_1:sel.2">Target</a> of <font color="Maroon" title="c7">G'</font>,the <a href="grcat_1.html#U3" title="GRCAT_1:sel.3">Fun</a> of <font color="Maroon" title="c7">G'</font> #) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="grcat_1.html#G1" title="GRCAT_1:aggr.1">GroupMorphismStr</a>(# <font color="Maroon" title="c5">G2'</font>,<font color="Maroon" title="c6">G3'</font>,<font color="Maroon" title="c12">g'</font> #)
 <b>and </b><br/><a NAME="E13:42_1_2"/>
<font color="Maroon" title="c12">g'</font> is <a href="grcat_1.html#V1" title="GRCAT_1:attr.1">additive</a>
 <b>by </b><i><a class="ref" href="grcat_1.html#T22" target="_self" title="GRCAT_1:th.22">Th22</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c13">f'</font> being   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c8">G1'</font>,<font color="Maroon" title="c5">G2'</font><b> such that </b><br/><a NAME="E15:42_1_2"/><i><font color="Green" title="E28">A14</font></i>: 
 <a href="grcat_1.html#G1" title="GRCAT_1:aggr.1">GroupMorphismStr</a>(# the <a href="grcat_1.html#U1" title="GRCAT_1:sel.1">Source</a> of <font color="Maroon" title="c11">F'</font>,the <a href="grcat_1.html#U2" title="GRCAT_1:sel.2">Target</a> of <font color="Maroon" title="c11">F'</font>,the <a href="grcat_1.html#U3" title="GRCAT_1:sel.3">Fun</a> of <font color="Maroon" title="c11">F'</font> #) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="grcat_1.html#G1" title="GRCAT_1:aggr.1">GroupMorphismStr</a>(# <font color="Maroon" title="c8">G1'</font>,<font color="Maroon" title="c5">G2'</font>,<font color="Maroon" title="c13">f'</font> #)
 <b>and </b><br/><a NAME="E16:42_1_2"/>
<font color="Maroon" title="c13">f'</font> is <a href="grcat_1.html#V1" title="GRCAT_1:attr.1">additive</a>
 <b>by </b><i><a class="ref" href="grcat_1.html#T22" target="_self" title="GRCAT_1:th.22">Th22</a></i>;<br/>
<b>thus </b><font color="Maroon" title="c3">S1</font> = 
 <a href="grcat_1.html#G1" title="GRCAT_1:aggr.1">GroupMorphismStr</a>(# <font color="Maroon" title="c8">G1'</font>,<font color="Maroon" title="c6">G3'</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c12">g'</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c13">f'</font></span>)</span> #)
<b>by </b><i><a class="txt" href="grcat_1.html#E1:42_1_2"><i><font color="Green" title="E22">A8</font></i></a>, <a class="txt" href="grcat_1.html#E12:42_1_2"><i><font color="Green" title="E27">A13</font></i></a>, <a class="txt" href="grcat_1.html#E15:42_1_2"><i><font color="Green" title="E28">A14</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c4">S2</font>
<b>by </b><i><a class="txt" href="grcat_1.html#E2:42_1_2"><i><font color="Green" title="E23">A9</font></i></a>, <a class="txt" href="grcat_1.html#E12:42_1_2"><i><font color="Green" title="E27">A13</font></i></a>, <a class="txt" href="grcat_1.html#E15:42_1_2"><i><font color="Green" title="E28">A14</font></i></a></i>
;<br/>


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