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<b>let </b><font color="Maroon" title="c1">UN</font> be   <a href="classes2.html#NM1" title="CLASSES2:NM.1">Universe</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">f</font>, <font color="Olive" title="b2">g</font> being  <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <span class="p1">(<span class="default"><a href="grcat_1.html#K27" title="GRCAT_1:func.27">GroupCat</a> <font color="Maroon" title="c1">UN</font></span>)</span><br/> for <font color="Olive" title="b3">f'</font>, <font color="Olive" title="b4">g'</font> being   <a href="grcat_1.html#M3" title="GRCAT_1:mode.3">Element</a> of  <a href="grcat_1.html#K19" title="GRCAT_1:func.19">Morphs</a> <span class="p1">(<span class="default"><a href="grcat_1.html#K18" title="GRCAT_1:func.18">GroupObjects</a> <font color="Maroon" title="c1">UN</font></span>)</span>  st <font color="Olive" title="b1">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">f'</font> &amp; <font color="Olive" title="b2">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">g'</font> holds <br/>( (  <a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <font color="Olive" title="b2">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <font color="Olive" title="b1">f</font> implies  <a href="grcat_1.html#K20" title="GRCAT_1:func.20">dom</a> <font color="Olive" title="b4">g'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="grcat_1.html#K21" title="GRCAT_1:func.21">cod</a> <font color="Olive" title="b3">f'</font> ) &amp; (  <a href="grcat_1.html#K20" title="GRCAT_1:func.20">dom</a> <font color="Olive" title="b4">g'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="grcat_1.html#K21" title="GRCAT_1:func.21">cod</a> <font color="Olive" title="b3">f'</font> implies  <a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <font color="Olive" title="b2">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <font color="Olive" title="b1">f</font> ) &amp; (  <a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <font color="Olive" title="b2">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <font color="Olive" title="b1">f</font> implies <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b4">g'</font>,<font color="Olive" title="b3">f'</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><a href="grcat_1.html#K26" title="GRCAT_1:func.26">comp</a> <span class="p2">(<span class="default"><a href="grcat_1.html#K18" title="GRCAT_1:func.18">GroupObjects</a> <font color="Maroon" title="c1">UN</font></span>)</span></span>)</span> ) &amp; ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b4">g'</font>,<font color="Olive" title="b3">f'</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><a href="grcat_1.html#K26" title="GRCAT_1:func.26">comp</a> <span class="p2">(<span class="default"><a href="grcat_1.html#K18" title="GRCAT_1:func.18">GroupObjects</a> <font color="Maroon" title="c1">UN</font></span>)</span></span>)</span> implies  <a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <font color="Olive" title="b2">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <font color="Olive" title="b1">f</font> ) &amp; (  <a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <font color="Olive" title="b2">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <font color="Olive" title="b1">f</font> implies <font color="Olive" title="b2">g</font> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <font color="Olive" title="b1">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">g'</font> <a href="grcat_1.html#K14" title="GRCAT_1:func.14">*</a> <font color="Olive" title="b3">f'</font> ) &amp; (  <a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <font color="Olive" title="b1">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <font color="Olive" title="b2">g</font> implies  <a href="grcat_1.html#K20" title="GRCAT_1:func.20">dom</a> <font color="Olive" title="b3">f'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="grcat_1.html#K20" title="GRCAT_1:func.20">dom</a> <font color="Olive" title="b4">g'</font> ) &amp; (  <a href="grcat_1.html#K20" title="GRCAT_1:func.20">dom</a> <font color="Olive" title="b3">f'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="grcat_1.html#K20" title="GRCAT_1:func.20">dom</a> <font color="Olive" title="b4">g'</font> implies  <a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <font color="Olive" title="b1">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <font color="Olive" title="b2">g</font> ) &amp; (  <a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <font color="Olive" title="b1">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <font color="Olive" title="b2">g</font> implies  <a href="grcat_1.html#K21" title="GRCAT_1:func.21">cod</a> <font color="Olive" title="b3">f'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="grcat_1.html#K21" title="GRCAT_1:func.21">cod</a> <font color="Olive" title="b4">g'</font> ) &amp; (  <a href="grcat_1.html#K21" title="GRCAT_1:func.21">cod</a> <font color="Olive" title="b3">f'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="grcat_1.html#K21" title="GRCAT_1:func.21">cod</a> <font color="Olive" title="b4">g'</font> implies  <a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <font color="Olive" title="b1">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <font color="Olive" title="b2">g</font> ) )</span><br/>

<b>set </b><font color="Maroon" title="c2">C</font> =  <a href="grcat_1.html#K27" title="GRCAT_1:func.27">GroupCat</a> <font color="Maroon" title="c1">UN</font>;<br/>
<b>set </b><font color="Maroon" title="c3">V</font> =  <a href="grcat_1.html#K18" title="GRCAT_1:func.18">GroupObjects</a> <font color="Maroon" title="c1">UN</font>;<br/>
<b>set </b><font color="Maroon" title="c4">X</font> =  <a href="grcat_1.html#K19" title="GRCAT_1:func.19">Morphs</a> <span class="p1">(<span class="default"><a href="grcat_1.html#K18" title="GRCAT_1:func.18">GroupObjects</a> <font color="Maroon" title="c1">UN</font></span>)</span>;<br/>
<b>let </b><font color="Maroon" title="c5">f</font>, <font color="Maroon" title="c6">g</font> be   <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <span class="p1">(<span class="default"><a href="grcat_1.html#K27" title="GRCAT_1:func.27">GroupCat</a> <font color="Maroon" title="c1">UN</font></span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">f'</font>, <font color="Olive" title="b2">g'</font> being   <a href="grcat_1.html#M3" title="GRCAT_1:mode.3">Element</a> of  <a href="grcat_1.html#K19" title="GRCAT_1:func.19">Morphs</a> <span class="p1">(<span class="default"><a href="grcat_1.html#K18" title="GRCAT_1:func.18">GroupObjects</a> <font color="Maroon" title="c1">UN</font></span>)</span>  st <font color="Maroon" title="c5">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">f'</font> &amp; <font color="Maroon" title="c6">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">g'</font> holds <br/>( (  <a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <font color="Maroon" title="c6">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <font color="Maroon" title="c5">f</font> implies  <a href="grcat_1.html#K20" title="GRCAT_1:func.20">dom</a> <font color="Olive" title="b2">g'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="grcat_1.html#K21" title="GRCAT_1:func.21">cod</a> <font color="Olive" title="b1">f'</font> ) &amp; (  <a href="grcat_1.html#K20" title="GRCAT_1:func.20">dom</a> <font color="Olive" title="b2">g'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="grcat_1.html#K21" title="GRCAT_1:func.21">cod</a> <font color="Olive" title="b1">f'</font> implies  <a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <font color="Maroon" title="c6">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <font color="Maroon" title="c5">f</font> ) &amp; (  <a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <font color="Maroon" title="c6">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <font color="Maroon" title="c5">f</font> implies <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b2">g'</font>,<font color="Olive" title="b1">f'</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><a href="grcat_1.html#K26" title="GRCAT_1:func.26">comp</a> <span class="p2">(<span class="default"><a href="grcat_1.html#K18" title="GRCAT_1:func.18">GroupObjects</a> <font color="Maroon" title="c1">UN</font></span>)</span></span>)</span> ) &amp; ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b2">g'</font>,<font color="Olive" title="b1">f'</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><a href="grcat_1.html#K26" title="GRCAT_1:func.26">comp</a> <span class="p2">(<span class="default"><a href="grcat_1.html#K18" title="GRCAT_1:func.18">GroupObjects</a> <font color="Maroon" title="c1">UN</font></span>)</span></span>)</span> implies  <a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <font color="Maroon" title="c6">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <font color="Maroon" title="c5">f</font> ) &amp; (  <a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <font color="Maroon" title="c6">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <font color="Maroon" title="c5">f</font> implies <font color="Maroon" title="c6">g</font> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <font color="Maroon" title="c5">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">g'</font> <a href="grcat_1.html#K14" title="GRCAT_1:func.14">*</a> <font color="Olive" title="b1">f'</font> ) &amp; (  <a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <font color="Maroon" title="c5">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <font color="Maroon" title="c6">g</font> implies  <a href="grcat_1.html#K20" title="GRCAT_1:func.20">dom</a> <font color="Olive" title="b1">f'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="grcat_1.html#K20" title="GRCAT_1:func.20">dom</a> <font color="Olive" title="b2">g'</font> ) &amp; (  <a href="grcat_1.html#K20" title="GRCAT_1:func.20">dom</a> <font color="Olive" title="b1">f'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="grcat_1.html#K20" title="GRCAT_1:func.20">dom</a> <font color="Olive" title="b2">g'</font> implies  <a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <font color="Maroon" title="c5">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <font color="Maroon" title="c6">g</font> ) &amp; (  <a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <font color="Maroon" title="c5">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <font color="Maroon" title="c6">g</font> implies  <a href="grcat_1.html#K21" title="GRCAT_1:func.21">cod</a> <font color="Olive" title="b1">f'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="grcat_1.html#K21" title="GRCAT_1:func.21">cod</a> <font color="Olive" title="b2">g'</font> ) &amp; (  <a href="grcat_1.html#K21" title="GRCAT_1:func.21">cod</a> <font color="Olive" title="b1">f'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="grcat_1.html#K21" title="GRCAT_1:func.21">cod</a> <font color="Olive" title="b2">g'</font> implies  <a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <font color="Maroon" title="c5">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <font color="Maroon" title="c6">g</font> ) )</span><br/><b>let </b><font color="Maroon" title="c7">f'</font>, <font color="Maroon" title="c8">g'</font> be    <a href="grcat_1.html#M3" title="GRCAT_1:mode.3">Element</a> of  <a href="grcat_1.html#K19" title="GRCAT_1:func.19">Morphs</a> <span class="p1">(<span class="default"><a href="grcat_1.html#K18" title="GRCAT_1:func.18">GroupObjects</a> <font color="Maroon" title="c1">UN</font></span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c5">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">f'</font> &amp; <font color="Maroon" title="c6">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">g'</font> implies ( (  <a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <font color="Maroon" title="c6">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <font color="Maroon" title="c5">f</font> implies  <a href="grcat_1.html#K20" title="GRCAT_1:func.20">dom</a> <font color="Maroon" title="c8">g'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="grcat_1.html#K21" title="GRCAT_1:func.21">cod</a> <font color="Maroon" title="c7">f'</font> ) &amp; (  <a href="grcat_1.html#K20" title="GRCAT_1:func.20">dom</a> <font color="Maroon" title="c8">g'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="grcat_1.html#K21" title="GRCAT_1:func.21">cod</a> <font color="Maroon" title="c7">f'</font> implies  <a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <font color="Maroon" title="c6">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <font color="Maroon" title="c5">f</font> ) &amp; (  <a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <font color="Maroon" title="c6">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <font color="Maroon" title="c5">f</font> implies <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c8">g'</font>,<font color="Maroon" title="c7">f'</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><a href="grcat_1.html#K26" title="GRCAT_1:func.26">comp</a> <span class="p2">(<span class="default"><a href="grcat_1.html#K18" title="GRCAT_1:func.18">GroupObjects</a> <font color="Maroon" title="c1">UN</font></span>)</span></span>)</span> ) &amp; ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c8">g'</font>,<font color="Maroon" title="c7">f'</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><a href="grcat_1.html#K26" title="GRCAT_1:func.26">comp</a> <span class="p2">(<span class="default"><a href="grcat_1.html#K18" title="GRCAT_1:func.18">GroupObjects</a> <font color="Maroon" title="c1">UN</font></span>)</span></span>)</span> implies  <a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <font color="Maroon" title="c6">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <font color="Maroon" title="c5">f</font> ) &amp; (  <a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <font color="Maroon" title="c6">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <font color="Maroon" title="c5">f</font> implies <font color="Maroon" title="c6">g</font> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <font color="Maroon" title="c5">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">g'</font> <a href="grcat_1.html#K14" title="GRCAT_1:func.14">*</a> <font color="Maroon" title="c7">f'</font> ) &amp; (  <a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <font color="Maroon" title="c5">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <font color="Maroon" title="c6">g</font> implies  <a href="grcat_1.html#K20" title="GRCAT_1:func.20">dom</a> <font color="Maroon" title="c7">f'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="grcat_1.html#K20" title="GRCAT_1:func.20">dom</a> <font color="Maroon" title="c8">g'</font> ) &amp; (  <a href="grcat_1.html#K20" title="GRCAT_1:func.20">dom</a> <font color="Maroon" title="c7">f'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="grcat_1.html#K20" title="GRCAT_1:func.20">dom</a> <font color="Maroon" title="c8">g'</font> implies  <a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <font color="Maroon" title="c5">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <font color="Maroon" title="c6">g</font> ) &amp; (  <a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <font color="Maroon" title="c5">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <font color="Maroon" title="c6">g</font> implies  <a href="grcat_1.html#K21" title="GRCAT_1:func.21">cod</a> <font color="Maroon" title="c7">f'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="grcat_1.html#K21" title="GRCAT_1:func.21">cod</a> <font color="Maroon" title="c8">g'</font> ) &amp; (  <a href="grcat_1.html#K21" title="GRCAT_1:func.21">cod</a> <font color="Maroon" title="c7">f'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="grcat_1.html#K21" title="GRCAT_1:func.21">cod</a> <font color="Maroon" title="c8">g'</font> implies  <a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <font color="Maroon" title="c5">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <font color="Maroon" title="c6">g</font> ) ) )</span><br/>



<b>assume </b><a NAME="E1:93"/><i><font color="Green" title="E53">A1</font></i>: 
( <font color="Maroon" title="c5">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">f'</font> &amp; <font color="Maroon" title="c6">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">g'</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( (  <a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <font color="Maroon" title="c6">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <font color="Maroon" title="c5">f</font> implies  <a href="grcat_1.html#K20" title="GRCAT_1:func.20">dom</a> <font color="Maroon" title="c8">g'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="grcat_1.html#K21" title="GRCAT_1:func.21">cod</a> <font color="Maroon" title="c7">f'</font> ) &amp; (  <a href="grcat_1.html#K20" title="GRCAT_1:func.20">dom</a> <font color="Maroon" title="c8">g'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="grcat_1.html#K21" title="GRCAT_1:func.21">cod</a> <font color="Maroon" title="c7">f'</font> implies  <a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <font color="Maroon" title="c6">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <font color="Maroon" title="c5">f</font> ) &amp; (  <a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <font color="Maroon" title="c6">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <font color="Maroon" title="c5">f</font> implies <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c8">g'</font>,<font color="Maroon" title="c7">f'</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><a href="grcat_1.html#K26" title="GRCAT_1:func.26">comp</a> <span class="p2">(<span class="default"><a href="grcat_1.html#K18" title="GRCAT_1:func.18">GroupObjects</a> <font color="Maroon" title="c1">UN</font></span>)</span></span>)</span> ) &amp; ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c8">g'</font>,<font color="Maroon" title="c7">f'</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><a href="grcat_1.html#K26" title="GRCAT_1:func.26">comp</a> <span class="p2">(<span class="default"><a href="grcat_1.html#K18" title="GRCAT_1:func.18">GroupObjects</a> <font color="Maroon" title="c1">UN</font></span>)</span></span>)</span> implies  <a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <font color="Maroon" title="c6">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <font color="Maroon" title="c5">f</font> ) &amp; (  <a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <font color="Maroon" title="c6">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <font color="Maroon" title="c5">f</font> implies <font color="Maroon" title="c6">g</font> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <font color="Maroon" title="c5">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">g'</font> <a href="grcat_1.html#K14" title="GRCAT_1:func.14">*</a> <font color="Maroon" title="c7">f'</font> ) &amp; (  <a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <font color="Maroon" title="c5">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <font color="Maroon" title="c6">g</font> implies  <a href="grcat_1.html#K20" title="GRCAT_1:func.20">dom</a> <font color="Maroon" title="c7">f'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="grcat_1.html#K20" title="GRCAT_1:func.20">dom</a> <font color="Maroon" title="c8">g'</font> ) &amp; (  <a href="grcat_1.html#K20" title="GRCAT_1:func.20">dom</a> <font color="Maroon" title="c7">f'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="grcat_1.html#K20" title="GRCAT_1:func.20">dom</a> <font color="Maroon" title="c8">g'</font> implies  <a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <font color="Maroon" title="c5">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <font color="Maroon" title="c6">g</font> ) &amp; (  <a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <font color="Maroon" title="c5">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <font color="Maroon" title="c6">g</font> implies  <a href="grcat_1.html#K21" title="GRCAT_1:func.21">cod</a> <font color="Maroon" title="c7">f'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="grcat_1.html#K21" title="GRCAT_1:func.21">cod</a> <font color="Maroon" title="c8">g'</font> ) &amp; (  <a href="grcat_1.html#K21" title="GRCAT_1:func.21">cod</a> <font color="Maroon" title="c7">f'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="grcat_1.html#K21" title="GRCAT_1:func.21">cod</a> <font color="Maroon" title="c8">g'</font> implies  <a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <font color="Maroon" title="c5">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <font color="Maroon" title="c6">g</font> ) )</span><br/>

<a NAME="E2:93"/><b>then </b><i><font color="Green" title="E54">A2</font></i>: 
 <a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <font color="Maroon" title="c6">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="grcat_1.html#K20" title="GRCAT_1:func.20">dom</a> <font color="Maroon" title="c8">g'</font>
 
<b>by </b><i><a class="ref" href="grcat_1.html#D32" target="_self" title="GRCAT_1:def.32">Def32</a></i>;<br/>
<a NAME="E3:93"/><i><font color="Green" title="E55">A3</font></i>: 
 <a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <font color="Maroon" title="c5">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="grcat_1.html#K21" title="GRCAT_1:func.21">cod</a> <font color="Maroon" title="c7">f'</font>
 
<b>by </b><i><a class="txt" href="#E1:93"><i><font color="Green" title="E53">A1</font></i></a>, <a class="ref" href="grcat_1.html#D33" target="_self" title="GRCAT_1:def.33">Def33</a></i>;<br/>
<a NAME="E4:93"/><i><font color="Green" title="E56">A4</font></i>: 
 <a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <font color="Maroon" title="c5">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="grcat_1.html#K20" title="GRCAT_1:func.20">dom</a> <font color="Maroon" title="c7">f'</font>
 
<b>by </b><i><a class="txt" href="#E1:93"><i><font color="Green" title="E53">A1</font></i></a>, <a class="ref" href="grcat_1.html#D32" target="_self" title="GRCAT_1:def.32">Def32</a></i>;<br/>
<a NAME="E5:93"/><i><font color="Green" title="E57">A5</font></i>: 
 <a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <font color="Maroon" title="c6">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="grcat_1.html#K21" title="GRCAT_1:func.21">cod</a> <font color="Maroon" title="c8">g'</font>
 
<b>by </b><i><a class="txt" href="#E1:93"><i><font color="Green" title="E53">A1</font></i></a>, <a class="ref" href="grcat_1.html#D33" target="_self" title="GRCAT_1:def.33">Def33</a></i>;<br/>
<b>thus </b><a NAME="E6:93"/>
(  <a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <font color="Maroon" title="c6">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <font color="Maroon" title="c5">f</font> iff  <a href="grcat_1.html#K20" title="GRCAT_1:func.20">dom</a> <font color="Maroon" title="c8">g'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="grcat_1.html#K21" title="GRCAT_1:func.21">cod</a> <font color="Maroon" title="c7">f'</font> )
 <b>by </b><i><a class="txt" href="#E1:93"><i><font color="Green" title="E53">A1</font></i></a>, <a class="txt" href="#E3:93"><i><font color="Green" title="E55">A3</font></i></a>, <a class="ref" href="grcat_1.html#D32" target="_self" title="GRCAT_1:def.32">Def32</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( (  <a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <font color="Maroon" title="c6">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <font color="Maroon" title="c5">f</font> implies <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c8">g'</font>,<font color="Maroon" title="c7">f'</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><a href="grcat_1.html#K26" title="GRCAT_1:func.26">comp</a> <span class="p2">(<span class="default"><a href="grcat_1.html#K18" title="GRCAT_1:func.18">GroupObjects</a> <font color="Maroon" title="c1">UN</font></span>)</span></span>)</span> ) &amp; ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c8">g'</font>,<font color="Maroon" title="c7">f'</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><a href="grcat_1.html#K26" title="GRCAT_1:func.26">comp</a> <span class="p2">(<span class="default"><a href="grcat_1.html#K18" title="GRCAT_1:func.18">GroupObjects</a> <font color="Maroon" title="c1">UN</font></span>)</span></span>)</span> implies  <a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <font color="Maroon" title="c6">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <font color="Maroon" title="c5">f</font> ) &amp; (  <a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <font color="Maroon" title="c6">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <font color="Maroon" title="c5">f</font> implies <font color="Maroon" title="c6">g</font> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <font color="Maroon" title="c5">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">g'</font> <a href="grcat_1.html#K14" title="GRCAT_1:func.14">*</a> <font color="Maroon" title="c7">f'</font> ) &amp; (  <a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <font color="Maroon" title="c5">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <font color="Maroon" title="c6">g</font> implies  <a href="grcat_1.html#K20" title="GRCAT_1:func.20">dom</a> <font color="Maroon" title="c7">f'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="grcat_1.html#K20" title="GRCAT_1:func.20">dom</a> <font color="Maroon" title="c8">g'</font> ) &amp; (  <a href="grcat_1.html#K20" title="GRCAT_1:func.20">dom</a> <font color="Maroon" title="c7">f'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="grcat_1.html#K20" title="GRCAT_1:func.20">dom</a> <font color="Maroon" title="c8">g'</font> implies  <a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <font color="Maroon" title="c5">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <font color="Maroon" title="c6">g</font> ) &amp; (  <a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <font color="Maroon" title="c5">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <font color="Maroon" title="c6">g</font> implies  <a href="grcat_1.html#K21" title="GRCAT_1:func.21">cod</a> <font color="Maroon" title="c7">f'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="grcat_1.html#K21" title="GRCAT_1:func.21">cod</a> <font color="Maroon" title="c8">g'</font> ) &amp; (  <a href="grcat_1.html#K21" title="GRCAT_1:func.21">cod</a> <font color="Maroon" title="c7">f'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="grcat_1.html#K21" title="GRCAT_1:func.21">cod</a> <font color="Maroon" title="c8">g'</font> implies  <a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <font color="Maroon" title="c5">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <font color="Maroon" title="c6">g</font> ) )</span><br/>

<b>thus </b><a NAME="E7:93"/><i><font color="Green" title="E58">A6</font></i>: 
(  <a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <font color="Maroon" title="c6">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <font color="Maroon" title="c5">f</font> iff <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c8">g'</font>,<font color="Maroon" title="c7">f'</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><a href="grcat_1.html#K26" title="GRCAT_1:func.26">comp</a> <span class="p2">(<span class="default"><a href="grcat_1.html#K18" title="GRCAT_1:func.18">GroupObjects</a> <font color="Maroon" title="c1">UN</font></span>)</span></span>)</span> )
 <b>by </b><i><a class="txt" href="#E2:93"><i><font color="Green" title="E54">A2</font></i></a>, <a class="txt" href="#E3:93"><i><font color="Green" title="E55">A3</font></i></a>, <a class="ref" href="grcat_1.html#D35" target="_self" title="GRCAT_1:def.35">Def35</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( (  <a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <font color="Maroon" title="c6">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <font color="Maroon" title="c5">f</font> implies <font color="Maroon" title="c6">g</font> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <font color="Maroon" title="c5">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">g'</font> <a href="grcat_1.html#K14" title="GRCAT_1:func.14">*</a> <font color="Maroon" title="c7">f'</font> ) &amp; (  <a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <font color="Maroon" title="c5">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <font color="Maroon" title="c6">g</font> implies  <a href="grcat_1.html#K20" title="GRCAT_1:func.20">dom</a> <font color="Maroon" title="c7">f'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="grcat_1.html#K20" title="GRCAT_1:func.20">dom</a> <font color="Maroon" title="c8">g'</font> ) &amp; (  <a href="grcat_1.html#K20" title="GRCAT_1:func.20">dom</a> <font color="Maroon" title="c7">f'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="grcat_1.html#K20" title="GRCAT_1:func.20">dom</a> <font color="Maroon" title="c8">g'</font> implies  <a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <font color="Maroon" title="c5">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <font color="Maroon" title="c6">g</font> ) &amp; (  <a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <font color="Maroon" title="c5">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <font color="Maroon" title="c6">g</font> implies  <a href="grcat_1.html#K21" title="GRCAT_1:func.21">cod</a> <font color="Maroon" title="c7">f'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="grcat_1.html#K21" title="GRCAT_1:func.21">cod</a> <font color="Maroon" title="c8">g'</font> ) &amp; (  <a href="grcat_1.html#K21" title="GRCAT_1:func.21">cod</a> <font color="Maroon" title="c7">f'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="grcat_1.html#K21" title="GRCAT_1:func.21">cod</a> <font color="Maroon" title="c8">g'</font> implies  <a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <font color="Maroon" title="c5">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <font color="Maroon" title="c6">g</font> ) )</span><br/>

<b>thus </b><a NAME="E8:93"/>
(  <a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <font color="Maroon" title="c6">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <font color="Maroon" title="c5">f</font> implies <font color="Maroon" title="c6">g</font> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <font color="Maroon" title="c5">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">g'</font> <a href="grcat_1.html#K14" title="GRCAT_1:func.14">*</a> <font color="Maroon" title="c7">f'</font> )
  <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( (  <a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <font color="Maroon" title="c5">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <font color="Maroon" title="c6">g</font> implies  <a href="grcat_1.html#K20" title="GRCAT_1:func.20">dom</a> <font color="Maroon" title="c7">f'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="grcat_1.html#K20" title="GRCAT_1:func.20">dom</a> <font color="Maroon" title="c8">g'</font> ) &amp; (  <a href="grcat_1.html#K20" title="GRCAT_1:func.20">dom</a> <font color="Maroon" title="c7">f'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="grcat_1.html#K20" title="GRCAT_1:func.20">dom</a> <font color="Maroon" title="c8">g'</font> implies  <a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <font color="Maroon" title="c5">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <font color="Maroon" title="c6">g</font> ) &amp; (  <a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <font color="Maroon" title="c5">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <font color="Maroon" title="c6">g</font> implies  <a href="grcat_1.html#K21" title="GRCAT_1:func.21">cod</a> <font color="Maroon" title="c7">f'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="grcat_1.html#K21" title="GRCAT_1:func.21">cod</a> <font color="Maroon" title="c8">g'</font> ) &amp; (  <a href="grcat_1.html#K21" title="GRCAT_1:func.21">cod</a> <font color="Maroon" title="c7">f'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="grcat_1.html#K21" title="GRCAT_1:func.21">cod</a> <font color="Maroon" title="c8">g'</font> implies  <a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <font color="Maroon" title="c5">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <font color="Maroon" title="c6">g</font> ) )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="93_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:93_1"/><i><font color="Green" title="E59">A7</font></i>: 
 <a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <font color="Maroon" title="c6">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <font color="Maroon" title="c5">f</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c6">g</font> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <font color="Maroon" title="c5">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">g'</font> <a href="grcat_1.html#K14" title="GRCAT_1:func.14">*</a> <font color="Maroon" title="c7">f'</font></span><br/>

<a NAME="E2:93_1"/><b>then </b>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c6">g</font>,<font color="Maroon" title="c5">f</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> the <a href="cat_1.html#U1" title="CAT_1:sel.1">Comp</a> of <span class="p1">(<span class="default"><a href="grcat_1.html#K27" title="GRCAT_1:func.27">GroupCat</a> <font color="Maroon" title="c1">UN</font></span>)</span>
 
<b>by </b><i><a class="ref" href="grcat_1.html#T46" target="_self" title="GRCAT_1:th.46">Th46</a></i>;<br/>
<b>hence </b><font color="Maroon" title="c6">g</font> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <font color="Maroon" title="c5">f</font> = 
<span class="p1">(<span class="default"><a href="grcat_1.html#K26" title="GRCAT_1:func.26">comp</a> <span class="p2">(<span class="default"><a href="grcat_1.html#K18" title="GRCAT_1:func.18">GroupObjects</a> <font color="Maroon" title="c1">UN</font></span>)</span></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c8">g'</font>,<font color="Maroon" title="c7">f'</font>
<b>by </b><i><a class="txt" href="#E1:93"><i><font color="Green" title="E53">A1</font></i></a>, <a class="ref" href="cat_1.html#D4" title="CAT_1:def.4">CAT_1:def 4</a></i>
<br/>.= 
<font color="Maroon" title="c8">g'</font> <a href="grcat_1.html#K14" title="GRCAT_1:func.14">*</a> <font color="Maroon" title="c7">f'</font>
<b>by </b><i><a class="txt" href="#E7:93"><i><font color="Green" title="E58">A6</font></i></a>, <a class="txt" href="#E1:93_1"><i><font color="Green" title="E59">A7</font></i></a>, <a class="ref" href="grcat_1.html#D35" target="_self" title="GRCAT_1:def.35">Def35</a></i>
;<br/> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>

<b>thus </b><a NAME="E9:93"/>
(  <a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <font color="Maroon" title="c5">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <font color="Maroon" title="c6">g</font> iff  <a href="grcat_1.html#K20" title="GRCAT_1:func.20">dom</a> <font color="Maroon" title="c7">f'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="grcat_1.html#K20" title="GRCAT_1:func.20">dom</a> <font color="Maroon" title="c8">g'</font> )
 <b>by </b><i><a class="txt" href="#E1:93"><i><font color="Green" title="E53">A1</font></i></a>, <a class="txt" href="#E4:93"><i><font color="Green" title="E56">A4</font></i></a>, <a class="ref" href="grcat_1.html#D32" target="_self" title="GRCAT_1:def.32">Def32</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  <a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <font color="Maroon" title="c5">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <font color="Maroon" title="c6">g</font> iff  <a href="grcat_1.html#K21" title="GRCAT_1:func.21">cod</a> <font color="Maroon" title="c7">f'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="grcat_1.html#K21" title="GRCAT_1:func.21">cod</a> <font color="Maroon" title="c8">g'</font> )</span><br/>

<b>thus </b><a NAME="E10:93"/>
(  <a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <font color="Maroon" title="c5">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <font color="Maroon" title="c6">g</font> iff  <a href="grcat_1.html#K21" title="GRCAT_1:func.21">cod</a> <font color="Maroon" title="c7">f'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="grcat_1.html#K21" title="GRCAT_1:func.21">cod</a> <font color="Maroon" title="c8">g'</font> )
 <b>by </b><i><a class="txt" href="#E1:93"><i><font color="Green" title="E53">A1</font></i></a>, <a class="txt" href="#E5:93"><i><font color="Green" title="E57">A5</font></i></a>, <a class="ref" href="grcat_1.html#D33" target="_self" title="GRCAT_1:def.33">Def33</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div>
