<?xml version="1.0"?>
<div class="add">

<span class="kw">let </span><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:()"><span class="comment"><font color="Red">::  thesis: </font></span></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/>





<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:31"/><span class="lab"><font color="Green" title="E18">A1</font></span>: 
<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>
 <span class="kw">and </span><br/><a NAME="E2:31"/><span class="lab"><font color="Green" title="E19">A2</font></span>: 
<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:()"><span class="comment"><font color="Red">::  thesis: </font></span></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/>

<span class="kw">consider </span><font color="Maroon" title="c4">F0</font> being   <a href="ringcat1.html#NM1" title="RINGCAT1:NM.1">RingMorphism</a><span class="kw"> such that </span><br/><a NAME="E4:31"/><span class="lab"><font color="Green" title="E20">A3</font></span>: 
 <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>
 <span class="kw">and </span><br/><a NAME="E5:31"/><span class="lab"><font color="Green" title="E21">A4</font></span>: 
 <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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:31"><span class="lab"><font color="Green" title="E18">A1</font></span></a>, <a class="ref" href="ringcat1.html#D8" target="_self" title="RINGCAT1:def.8">Def8</a></span>;<br/>
<span class="kw">reconsider </span><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> <span class="kw">by</span> <span class="lab"><a class="txt" href="ringcat1.html#E13"><span class="lab"><font color="Green" title="E6">Lm2</font></span></a></span>;<br/>
<a NAME="E7:31"/>
(  <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> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:31"><span class="lab"><font color="Green" title="E20">A3</font></span></a>, <a class="txt" href="#E5:31"><span class="lab"><font color="Green" title="E21">A4</font></span></a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c6">F9</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> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:31"><span class="lab"><font color="Green" title="E18">A1</font></span></a>, <a class="ref" href="ringcat1.html#D9" target="_self" title="RINGCAT1:def.9">Def9</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c7">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><span class="kw"> such that </span><br/><a NAME="E10:31"/><span class="lab"><font color="Green" title="E22">A5</font></span>: 
<font color="Maroon" title="c6">F9</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="c7">f</font> #)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:31"><span class="lab"><font color="Green" title="E18">A1</font></span></a>, <a class="txt" href="ringcat1.html#E23"><span class="lab"><font color="Green" title="E15">Lm8</font></span></a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c8">G0</font> being   <a href="ringcat1.html#NM1" title="RINGCAT1:NM.1">RingMorphism</a><span class="kw"> such that </span><br/><a NAME="E12:31"/><span class="lab"><font color="Green" title="E23">A6</font></span>: 
 <a href="ringcat1.html#K1" title="RINGCAT1:func.1">dom</a> <font color="Maroon" title="c8">G0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">G2</font>
 <span class="kw">and </span><br/><a NAME="E13:31"/><span class="lab"><font color="Green" title="E24">A7</font></span>: 
 <a href="ringcat1.html#K2" title="RINGCAT1:func.2">cod</a> <font color="Maroon" title="c8">G0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">G3</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:31"><span class="lab"><font color="Green" title="E19">A2</font></span></a>, <a class="ref" href="ringcat1.html#D8" target="_self" title="RINGCAT1:def.8">Def8</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c9">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="c8">G0</font>,the <a href="ringcat1.html#U2" title="RINGCAT1:sel.2">Cod</a> of <font color="Maroon" title="c8">G0</font>,the <a href="ringcat1.html#U3" title="RINGCAT1:sel.3">Fun</a> of <font color="Maroon" title="c8">G0</font> #) as   <a href="ringcat1.html#NM1" title="RINGCAT1:NM.1">RingMorphism</a> <span class="kw">by</span> <span class="lab"><a class="txt" href="ringcat1.html#E13"><span class="lab"><font color="Green" title="E6">Lm2</font></span></a></span>;<br/>
<a NAME="E15:31"/>
(  <a href="ringcat1.html#K1" title="RINGCAT1:func.1">dom</a> <font color="Maroon" title="c9">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="c9">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">G3</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:31"><span class="lab"><font color="Green" title="E23">A6</font></span></a>, <a class="txt" href="#E13:31"><span class="lab"><font color="Green" title="E24">A7</font></span></a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c10">G9</font> = <font color="Maroon" title="c9">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> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:31"><span class="lab"><font color="Green" title="E19">A2</font></span></a>, <a class="ref" href="ringcat1.html#D9" target="_self" title="RINGCAT1:def.9">Def9</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c11">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><span class="kw"> such that </span><br/><a NAME="E18:31"/><span class="lab"><font color="Green" title="E25">A8</font></span>: 
<font color="Maroon" title="c10">G9</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="c11">g</font> #)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:31"><span class="lab"><font color="Green" title="E19">A2</font></span></a>, <a class="txt" href="ringcat1.html#E23"><span class="lab"><font color="Green" title="E15">Lm8</font></span></a></span>;<br/>
<a NAME="E19:31"/>
 <a href="ringcat1.html#K1" title="RINGCAT1:func.1">dom</a> <font color="Maroon" title="c9">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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:31"><span class="lab"><font color="Green" title="E21">A4</font></span></a>, <a class="txt" href="#E12:31"><span class="lab"><font color="Green" title="E23">A6</font></span></a></span>;<br/>
<a NAME="E20:31"/><span class="kw">then </span>
<font color="Maroon" title="c9">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="c11">g</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c7">f</font></span>)</span> #)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:31"><span class="lab"><font color="Green" title="E25">A8</font></span></a>, <a class="txt" href="#E10:31"><span class="lab"><font color="Green" title="E22">A5</font></span></a>, <a class="ref" href="ringcat1.html#D10" target="_self" title="RINGCAT1:def.10">Def10</a></span>;<br/>
<a NAME="E21:31"/><span class="kw">then </span>
(  <a href="ringcat1.html#K1" title="RINGCAT1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">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="c9">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/>
<span class="kw">hence </span><a NAME="E22:31"/>
<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 class="kw">by</span> <span class="lab"><a class="ref" href="ringcat1.html#D8" target="_self" title="RINGCAT1:def.8">Def8</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
