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<span class="kw">defpred </span><font color="Maroon">S<sub>1</sub></font>[  <a href="grcat_1.html#NM1" title="GRCAT_1:NM.1">GroupMorphism</a>,  <a href="grcat_1.html#NM1" title="GRCAT_1:NM.1">GroupMorphism</a>] <span class="kw">means </span> <a href="grcat_1.html#K8" title="GRCAT_1:func.8">dom</a> $1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="grcat_1.html#K9" title="GRCAT_1:func.9">cod</a> $2;<br/>
<span class="kw">let </span><font color="Maroon" title="c1">g</font>, <font color="Maroon" title="c2">f</font> be   <a href="grcat_1.html#NM1" title="GRCAT_1:NM.1">GroupMorphism</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  <a href="grcat_1.html#K8" title="GRCAT_1:func.8">dom</a> <font color="Maroon" title="c1">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="grcat_1.html#K9" title="GRCAT_1:func.9">cod</a> <font color="Maroon" title="c2">f</font> implies  ex <font color="Olive" title="b1">G1</font>, <font color="Olive" title="b2">G2</font>, <font color="Olive" title="b3">G3</font> being   <a href="vectsp_1.html#NM1" title="VECTSP_1:NM.1">AddGroup</a> st <br/>( <font color="Maroon" title="c1">g</font> is    <a href="grcat_1.html#M1" title="GRCAT_1:mode.1">Morphism</a> of <font color="Olive" title="b2">G2</font>,<font color="Olive" title="b3">G3</font> &amp; <font color="Maroon" title="c2">f</font> is    <a href="grcat_1.html#M1" title="GRCAT_1:mode.1">Morphism</a> of <font color="Olive" title="b1">G1</font>,<font color="Olive" title="b2">G2</font> ) )</span><br/>

<span class="kw">assume </span><a NAME="E1:39"/><span class="lab"><font color="Green" title="E16">A1</font></span>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c1">g</font>,<font color="Maroon" title="c2">f</font>]
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">G1</font>, <font color="Olive" title="b2">G2</font>, <font color="Olive" title="b3">G3</font> being   <a href="vectsp_1.html#NM1" title="VECTSP_1:NM.1">AddGroup</a> st <br/>( <font color="Maroon" title="c1">g</font> is    <a href="grcat_1.html#M1" title="GRCAT_1:mode.1">Morphism</a> of <font color="Olive" title="b2">G2</font>,<font color="Olive" title="b3">G3</font> &amp; <font color="Maroon" title="c2">f</font> is    <a href="grcat_1.html#M1" title="GRCAT_1:mode.1">Morphism</a> of <font color="Olive" title="b1">G1</font>,<font color="Olive" title="b2">G2</font> )</span><br/>

<span class="kw">consider </span><font color="Maroon" title="c3">G2</font>, <font color="Maroon" title="c4">G3</font> being   <a href="vectsp_1.html#NM1" title="VECTSP_1:NM.1">AddGroup</a><span class="kw"> such that </span><br/><a NAME="E3:39"/><span class="lab"><font color="Green" title="E17">A2</font></span>: 
<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="c3">G2</font>,<font color="Maroon" title="c4">G3</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="grcat_1.html#T14" target="_self" title="GRCAT_1:th.14">Th24</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c5">G1</font>, <font color="Maroon" title="c6">G29</font> being   <a href="vectsp_1.html#NM1" title="VECTSP_1:NM.1">AddGroup</a><span class="kw"> such that </span><br/><a NAME="E5:39"/><span class="lab"><font color="Green" title="E18">A3</font></span>: 
<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="c5">G1</font>,<font color="Maroon" title="c6">G29</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="grcat_1.html#T14" target="_self" title="GRCAT_1:th.14">Th24</a></span>;<br/>
<a NAME="E6:39"/><span class="lab"><font color="Green" title="E19">A4</font></span>: 
<font color="Maroon" title="c6">G29</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="grcat_1.html#K9" title="GRCAT_1:func.9">cod</a> <font color="Maroon" title="c2">f</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:39"><span class="lab"><font color="Green" title="E18">A3</font></span></a>, <a class="ref" href="grcat_1.html#D14" target="_self" title="GRCAT_1:def.14">Def19</a></span>;<br/>
<a NAME="E7:39"/>
<font color="Maroon" title="c3">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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:39"><span class="lab"><font color="Green" title="E17">A2</font></span></a>, <a class="ref" href="grcat_1.html#D14" target="_self" title="GRCAT_1:def.14">Def19</a></span>;<br/>
<span class="kw">hence </span><a NAME="E8:39"/>
 ex <font color="Olive" title="b1">G1</font>, <font color="Olive" title="b2">G2</font>, <font color="Olive" title="b3">G3</font> being   <a href="vectsp_1.html#NM1" title="VECTSP_1:NM.1">AddGroup</a> st <br/>( <font color="Maroon" title="c1">g</font> is    <a href="grcat_1.html#M1" title="GRCAT_1:mode.1">Morphism</a> of <font color="Olive" title="b2">G2</font>,<font color="Olive" title="b3">G3</font> &amp; <font color="Maroon" title="c2">f</font> is    <a href="grcat_1.html#M1" title="GRCAT_1:mode.1">Morphism</a> of <font color="Olive" title="b1">G1</font>,<font color="Olive" title="b2">G2</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:39"><span class="lab"><font color="Green" title="E16">A1</font></span></a>, <a class="txt" href="#E3:39"><span class="lab"><font color="Green" title="E17">A2</font></span></a>, <a class="txt" href="#E5:39"><span class="lab"><font color="Green" title="E18">A3</font></span></a>, <a class="txt" href="#E6:39"><span class="lab"><font color="Green" title="E19">A4</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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