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<b>let </b><font color="Maroon" title="c1">G1</font>, <font color="Maroon" title="c2">G2</font>, <font color="Maroon" title="c3">G3</font> be   <a href="funcsdom.html#NM1" title="FUNCSDOM:NM.1">Ring</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c1">G1</font> <a href="ringcat1.html#R1" title="RINGCAT1:pred.1">&lt;=</a> <font color="Maroon" title="c2">G2</font> &amp; <font color="Maroon" title="c2">G2</font> <a href="ringcat1.html#R1" title="RINGCAT1:pred.1">&lt;=</a> <font color="Maroon" title="c3">G3</font> implies <font color="Maroon" title="c1">G1</font> <a href="ringcat1.html#R1" title="RINGCAT1:pred.1">&lt;=</a> <font color="Maroon" title="c3">G3</font> )</span><br/>





<b>assume </b><a NAME="E1:28"/><i><font color="Green" title="E18">A1</font></i>: 
( <font color="Maroon" title="c1">G1</font> <a href="ringcat1.html#R1" title="RINGCAT1:pred.1">&lt;=</a> <font color="Maroon" title="c2">G2</font> &amp; <font color="Maroon" title="c2">G2</font> <a href="ringcat1.html#R1" title="RINGCAT1:pred.1">&lt;=</a> <font color="Maroon" title="c3">G3</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c1">G1</font> <a href="ringcat1.html#R1" title="RINGCAT1:pred.1">&lt;=</a> <font color="Maroon" title="c3">G3</font></span><br/>

<b>then consider </b><font color="Maroon" title="c4">F0</font> being   <a href="ringcat1.html#NM1" title="RINGCAT1:NM.1">RingMorphism</a><b> such that </b><br/><a NAME="E3:28"/><i><font color="Green" title="E19">A2</font></i>: 
(  <a href="ringcat1.html#K1" title="RINGCAT1:func.1">dom</a> <font color="Maroon" title="c4">F0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">G1</font> &amp;  <a href="ringcat1.html#K2" title="RINGCAT1:func.2">cod</a> <font color="Maroon" title="c4">F0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">G2</font> )
 <b>by </b><i><a class="ref" href="ringcat1.html#D8" target="_self" title="RINGCAT1:def.8">Def8</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c5">F</font> =  <a href="ringcat1.html#G1" title="RINGCAT1:aggr.1">RingMorphismStr</a>(# the <a href="ringcat1.html#U1" title="RINGCAT1:sel.1">Dom</a> of <font color="Maroon" title="c4">F0</font>,the <a href="ringcat1.html#U2" title="RINGCAT1:sel.2">Cod</a> of <font color="Maroon" title="c4">F0</font>,the <a href="ringcat1.html#U3" title="RINGCAT1:sel.3">Fun</a> of <font color="Maroon" title="c4">F0</font> #) as   <a href="ringcat1.html#NM1" title="RINGCAT1:NM.1">RingMorphism</a> <b>by </b><i><a class="txt" href="ringcat1.html#E13"><i><font color="Green" title="E6">Lm2</font></i></a></i>;<br/>
<a NAME="E5:28"/>
(  <a href="ringcat1.html#K1" title="RINGCAT1:func.1">dom</a> <font color="Maroon" title="c5">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">G1</font> &amp;  <a href="ringcat1.html#K2" title="RINGCAT1:func.2">cod</a> <font color="Maroon" title="c5">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">G2</font> )
 
<b>by </b><i><a class="txt" href="ringcat1.html#E3:28"><i><font color="Green" title="E19">A2</font></i></a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c6">F'</font> = <font color="Maroon" title="c5">F</font> as    <a href="ringcat1.html#M1" title="RINGCAT1:mode.1">Morphism</a> of <font color="Maroon" title="c1">G1</font>,<font color="Maroon" title="c2">G2</font> <b>by </b><i><a class="txt" href="ringcat1.html#E1:28"><i><font color="Green" title="E18">A1</font></i></a>, <a class="ref" href="ringcat1.html#D9" target="_self" title="RINGCAT1:def.9">Def9</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c7">G0</font> being   <a href="ringcat1.html#NM1" title="RINGCAT1:NM.1">RingMorphism</a><b> such that </b><br/><a NAME="E8:28"/><i><font color="Green" title="E20">A3</font></i>: 
(  <a href="ringcat1.html#K1" title="RINGCAT1:func.1">dom</a> <font color="Maroon" title="c7">G0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">G2</font> &amp;  <a href="ringcat1.html#K2" title="RINGCAT1:func.2">cod</a> <font color="Maroon" title="c7">G0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">G3</font> )
 <b>by </b><i><a class="txt" href="ringcat1.html#E1:28"><i><font color="Green" title="E18">A1</font></i></a>, <a class="ref" href="ringcat1.html#D8" target="_self" title="RINGCAT1:def.8">Def8</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c8">G</font> =  <a href="ringcat1.html#G1" title="RINGCAT1:aggr.1">RingMorphismStr</a>(# the <a href="ringcat1.html#U1" title="RINGCAT1:sel.1">Dom</a> of <font color="Maroon" title="c7">G0</font>,the <a href="ringcat1.html#U2" title="RINGCAT1:sel.2">Cod</a> of <font color="Maroon" title="c7">G0</font>,the <a href="ringcat1.html#U3" title="RINGCAT1:sel.3">Fun</a> of <font color="Maroon" title="c7">G0</font> #) as   <a href="ringcat1.html#NM1" title="RINGCAT1:NM.1">RingMorphism</a> <b>by </b><i><a class="txt" href="ringcat1.html#E13"><i><font color="Green" title="E6">Lm2</font></i></a></i>;<br/>
<a NAME="E10:28"/>
(  <a href="ringcat1.html#K1" title="RINGCAT1:func.1">dom</a> <font color="Maroon" title="c8">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">G2</font> &amp;  <a href="ringcat1.html#K2" title="RINGCAT1:func.2">cod</a> <font color="Maroon" title="c8">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">G3</font> )
 
<b>by </b><i><a class="txt" href="ringcat1.html#E8:28"><i><font color="Green" title="E20">A3</font></i></a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c9">G'</font> = <font color="Maroon" title="c8">G</font> as    <a href="ringcat1.html#M1" title="RINGCAT1:mode.1">Morphism</a> of <font color="Maroon" title="c2">G2</font>,<font color="Maroon" title="c3">G3</font> <b>by </b><i><a class="txt" href="ringcat1.html#E1:28"><i><font color="Green" title="E18">A1</font></i></a>, <a class="ref" href="ringcat1.html#D9" target="_self" title="RINGCAT1:def.9">Def9</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c10">g</font> being   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c2">G2</font>,<font color="Maroon" title="c3">G3</font><b> such that </b><br/><a NAME="E13:28"/><i><font color="Green" title="E21">A4</font></i>: 
<font color="Maroon" title="c9">G'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ringcat1.html#G1" title="RINGCAT1:aggr.1">RingMorphismStr</a>(# <font color="Maroon" title="c2">G2</font>,<font color="Maroon" title="c3">G3</font>,<font color="Maroon" title="c10">g</font> #)
 <b>by </b><i><a class="txt" href="ringcat1.html#E1:28"><i><font color="Green" title="E18">A1</font></i></a>, <a class="txt" href="ringcat1.html#E23"><i><font color="Green" title="E15">Lm8</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon" title="c11">f</font> being   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c1">G1</font>,<font color="Maroon" title="c2">G2</font><b> such that </b><br/><a NAME="E15:28"/><i><font color="Green" title="E22">A5</font></i>: 
<font color="Maroon" title="c6">F'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ringcat1.html#G1" title="RINGCAT1:aggr.1">RingMorphismStr</a>(# <font color="Maroon" title="c1">G1</font>,<font color="Maroon" title="c2">G2</font>,<font color="Maroon" title="c11">f</font> #)
 <b>by </b><i><a class="txt" href="ringcat1.html#E1:28"><i><font color="Green" title="E18">A1</font></i></a>, <a class="txt" href="ringcat1.html#E23"><i><font color="Green" title="E15">Lm8</font></i></a></i>;<br/>
<a NAME="E16:28"/>
 <a href="ringcat1.html#K1" title="RINGCAT1:func.1">dom</a> <font color="Maroon" title="c8">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ringcat1.html#K2" title="RINGCAT1:func.2">cod</a> <font color="Maroon" title="c5">F</font>
 
<b>by </b><i><a class="txt" href="ringcat1.html#E3:28"><i><font color="Green" title="E19">A2</font></i></a>, <a class="txt" href="ringcat1.html#E8:28"><i><font color="Green" title="E20">A3</font></i></a></i>;<br/>
<a NAME="E17:28"/><b>then </b>
<font color="Maroon" title="c8">G</font> <a href="ringcat1.html#K6" title="RINGCAT1:func.6">*</a> <font color="Maroon" title="c5">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ringcat1.html#G1" title="RINGCAT1:aggr.1">RingMorphismStr</a>(# <font color="Maroon" title="c1">G1</font>,<font color="Maroon" title="c3">G3</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c10">g</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c11">f</font></span>)</span> #)
 
<b>by </b><i><a class="txt" href="ringcat1.html#E13:28"><i><font color="Green" title="E21">A4</font></i></a>, <a class="txt" href="ringcat1.html#E15:28"><i><font color="Green" title="E22">A5</font></i></a>, <a class="ref" href="ringcat1.html#D10" target="_self" title="RINGCAT1:def.10">Def10</a></i>;<br/>
<a NAME="E18:28"/><b>then </b>
(  <a href="ringcat1.html#K1" title="RINGCAT1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">G</font> <a href="ringcat1.html#K6" title="RINGCAT1:func.6">*</a> <font color="Maroon" title="c5">F</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">G1</font> &amp;  <a href="ringcat1.html#K2" title="RINGCAT1:func.2">cod</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">G</font> <a href="ringcat1.html#K6" title="RINGCAT1:func.6">*</a> <font color="Maroon" title="c5">F</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">G3</font> )
 
;<br/>
<b>hence </b><a NAME="E19:28"/>
<font color="Maroon" title="c1">G1</font> <a href="ringcat1.html#R1" title="RINGCAT1:pred.1">&lt;=</a> <font color="Maroon" title="c3">G3</font>
 <b>by </b><i><a class="ref" href="ringcat1.html#D8" target="_self" title="RINGCAT1:def.8">Def8</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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