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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>, <font color="Maroon" title="c4">G4</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"> for <font color="Olive" title="b1">f</font> being   <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><br/> for <font color="Olive" title="b2">g</font> being   <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><br/> for <font color="Olive" title="b3">h</font> being   <a href="ringcat1.html#M1" title="RINGCAT1:mode.1">Morphism</a> of <font color="Maroon" title="c3">G3</font>,<font color="Maroon" title="c4">G4</font>  st <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> &amp; <font color="Maroon" title="c3">G3</font> <a href="ringcat1.html#R1" title="RINGCAT1:pred.1">&lt;=</a> <font color="Maroon" title="c4">G4</font> holds <br/><font color="Olive" title="b3">h</font> <a href="ringcat1.html#K6" title="RINGCAT1:func.6">*</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">g</font> <a href="ringcat1.html#K6" title="RINGCAT1:func.6">*</a> <font color="Olive" title="b1">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b3">h</font> <a href="ringcat1.html#K6" title="RINGCAT1:func.6">*</a> <font color="Olive" title="b2">g</font></span>)</span> <a href="ringcat1.html#K6" title="RINGCAT1:func.6">*</a> <font color="Olive" title="b1">f</font></span><br/><b>let </b><font color="Maroon" title="c5">f</font> be    <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>; <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">g</font> being   <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><br/> for <font color="Olive" title="b2">h</font> being   <a href="ringcat1.html#M1" title="RINGCAT1:mode.1">Morphism</a> of <font color="Maroon" title="c3">G3</font>,<font color="Maroon" title="c4">G4</font>  st <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> &amp; <font color="Maroon" title="c3">G3</font> <a href="ringcat1.html#R1" title="RINGCAT1:pred.1">&lt;=</a> <font color="Maroon" title="c4">G4</font> holds <br/><font color="Olive" title="b2">h</font> <a href="ringcat1.html#K6" title="RINGCAT1:func.6">*</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">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> <span class="p1">(<span class="default"><font color="Olive" title="b2">h</font> <a href="ringcat1.html#K6" title="RINGCAT1:func.6">*</a> <font color="Olive" title="b1">g</font></span>)</span> <a href="ringcat1.html#K6" title="RINGCAT1:func.6">*</a> <font color="Maroon" title="c5">f</font></span><br/><b>let </b><font color="Maroon" title="c6">g</font> be    <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>; <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">h</font> being   <a href="ringcat1.html#M1" title="RINGCAT1:mode.1">Morphism</a> of <font color="Maroon" title="c3">G3</font>,<font color="Maroon" title="c4">G4</font>  st <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> &amp; <font color="Maroon" title="c3">G3</font> <a href="ringcat1.html#R1" title="RINGCAT1:pred.1">&lt;=</a> <font color="Maroon" title="c4">G4</font> holds <br/><font color="Olive" title="b1">h</font> <a href="ringcat1.html#K6" title="RINGCAT1:func.6">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">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> <span class="p1">(<span class="default"><font color="Olive" title="b1">h</font> <a href="ringcat1.html#K6" title="RINGCAT1:func.6">*</a> <font color="Maroon" title="c6">g</font></span>)</span> <a href="ringcat1.html#K6" title="RINGCAT1:func.6">*</a> <font color="Maroon" title="c5">f</font></span><br/><b>let </b><font color="Maroon" title="c7">h</font> be    <a href="ringcat1.html#M1" title="RINGCAT1:mode.1">Morphism</a> of <font color="Maroon" title="c3">G3</font>,<font color="Maroon" title="c4">G4</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="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> &amp; <font color="Maroon" title="c3">G3</font> <a href="ringcat1.html#R1" title="RINGCAT1:pred.1">&lt;=</a> <font color="Maroon" title="c4">G4</font> implies <font color="Maroon" title="c7">h</font> <a href="ringcat1.html#K6" title="RINGCAT1:func.6">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">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> <span class="p1">(<span class="default"><font color="Maroon" title="c7">h</font> <a href="ringcat1.html#K6" title="RINGCAT1:func.6">*</a> <font color="Maroon" title="c6">g</font></span>)</span> <a href="ringcat1.html#K6" title="RINGCAT1:func.6">*</a> <font color="Maroon" title="c5">f</font> )</span><br/>













<b>assume </b><a NAME="E1:33"/><i><font color="Green" title="E23">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> &amp; <font color="Maroon" title="c3">G3</font> <a href="ringcat1.html#R1" title="RINGCAT1:pred.1">&lt;=</a> <font color="Maroon" title="c4">G4</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c7">h</font> <a href="ringcat1.html#K6" title="RINGCAT1:func.6">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">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> <span class="p1">(<span class="default"><font color="Maroon" title="c7">h</font> <a href="ringcat1.html#K6" title="RINGCAT1:func.6">*</a> <font color="Maroon" title="c6">g</font></span>)</span> <a href="ringcat1.html#K6" title="RINGCAT1:func.6">*</a> <font color="Maroon" title="c5">f</font></span><br/>

<b>then consider </b><font color="Maroon" title="c8">f0</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="E3:33"/><i><font color="Green" title="E24">A2</font></i>: 
<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="c2">G2</font>,<font color="Maroon" title="c8">f0</font> #)
 <b>by </b><i><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="c9">g0</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="E5:33"/><i><font color="Green" title="E25">A3</font></i>: 
<font color="Maroon" title="c6">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="c9">g0</font> #)
 <b>by </b><i><a class="txt" href="#E1:33"><i><font color="Green" title="E23">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="c10">h0</font> being   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c3">G3</font>,<font color="Maroon" title="c4">G4</font><b> such that </b><br/><a NAME="E7:33"/><i><font color="Green" title="E26">A4</font></i>: 
<font color="Maroon" title="c7">h</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="c3">G3</font>,<font color="Maroon" title="c4">G4</font>,<font color="Maroon" title="c10">h0</font> #)
 <b>by </b><i><a class="txt" href="#E1:33"><i><font color="Green" title="E23">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="E8:33"/><i><font color="Green" title="E27">A5</font></i>: 
(  <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> &amp;  <a href="ringcat1.html#K1" title="RINGCAT1:func.1">dom</a> <font color="Maroon" title="c6">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="c6">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">G3</font> &amp;  <a href="ringcat1.html#K1" title="RINGCAT1:func.1">dom</a> <font color="Maroon" title="c7">h</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="#E3:33"><i><font color="Green" title="E24">A2</font></i></a>, <a class="txt" href="#E5:33"><i><font color="Green" title="E25">A3</font></i></a>, <a class="txt" href="#E7:33"><i><font color="Green" title="E26">A4</font></i></a></i>;<br/>
<a NAME="E9:33"/><b>then </b><i><font color="Green" title="E28">A6</font></i>: 
<font color="Maroon" title="c6">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="c9">g0</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c8">f0</font></span>)</span> #)
 
<b>by </b><i><a class="txt" href="#E3:33"><i><font color="Green" title="E24">A2</font></i></a>, <a class="txt" href="#E5:33"><i><font color="Green" title="E25">A3</font></i></a>, <a class="ref" href="ringcat1.html#D10" target="_self" title="RINGCAT1:def.10">Def10</a></i>;<br/>
<a NAME="E10:33"/><i><font color="Green" title="E29">A7</font></i>: 
<font color="Maroon" title="c7">h</font> <a href="ringcat1.html#K6" title="RINGCAT1:func.6">*</a> <font color="Maroon" title="c6">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="c4">G4</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c10">h0</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c9">g0</font></span>)</span> #)
 
<b>by </b><i><a class="txt" href="#E5:33"><i><font color="Green" title="E25">A3</font></i></a>, <a class="txt" href="#E7:33"><i><font color="Green" title="E26">A4</font></i></a>, <a class="txt" href="#E8:33"><i><font color="Green" title="E27">A5</font></i></a>, <a class="ref" href="ringcat1.html#D10" target="_self" title="RINGCAT1:def.10">Def10</a></i>;<br/>
<a NAME="E11:33"/><i><font color="Green" title="E30">A8</font></i>: 
(  <a href="ringcat1.html#K2" title="RINGCAT1:func.2">cod</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">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> &amp;  <a href="ringcat1.html#K1" title="RINGCAT1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">h</font> <a href="ringcat1.html#K6" title="RINGCAT1:func.6">*</a> <font color="Maroon" title="c6">g</font></span>)</span> <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="#E8:33"><i><font color="Green" title="E27">A5</font></i></a>, <a class="ref" href="ringcat1.html#T11" target="_self" title="RINGCAT1:th.11">Th11</a></i>;<br/>
<b>then </b><font color="Maroon" title="c7">h</font> <a href="ringcat1.html#K6" title="RINGCAT1:func.6">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">g</font> <a href="ringcat1.html#K6" title="RINGCAT1:func.6">*</a> <font color="Maroon" title="c5">f</font></span>)</span> = 
 <a href="ringcat1.html#G1" title="RINGCAT1:aggr.1">RingMorphismStr</a>(# <font color="Maroon" title="c1">G1</font>,<font color="Maroon" title="c4">G4</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c10">h0</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">g0</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c8">f0</font></span>)</span></span>)</span> #)
<b>by </b><i><a class="txt" href="#E7:33"><i><font color="Green" title="E26">A4</font></i></a>, <a class="txt" href="#E8:33"><i><font color="Green" title="E27">A5</font></i></a>, <a class="txt" href="#E9:33"><i><font color="Green" title="E28">A6</font></i></a>, <a class="ref" href="ringcat1.html#D10" target="_self" title="RINGCAT1:def.10">Def10</a></i>
<br/>.= 
 <a href="ringcat1.html#G1" title="RINGCAT1:aggr.1">RingMorphismStr</a>(# <font color="Maroon" title="c1">G1</font>,<font color="Maroon" title="c4">G4</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c10">h0</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c9">g0</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c8">f0</font></span>)</span> #)
<b>by </b><i><a class="ref" href="relat_1.html#T55" title="RELAT_1:th.55">RELAT_1:55</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c7">h</font> <a href="ringcat1.html#K6" title="RINGCAT1:func.6">*</a> <font color="Maroon" title="c6">g</font></span>)</span> <a href="ringcat1.html#K6" title="RINGCAT1:func.6">*</a> <font color="Maroon" title="c5">f</font>
<b>by </b><i><a class="txt" href="#E3:33"><i><font color="Green" title="E24">A2</font></i></a>, <a class="txt" href="#E8:33"><i><font color="Green" title="E27">A5</font></i></a>, <a class="txt" href="#E10:33"><i><font color="Green" title="E29">A7</font></i></a>, <a class="txt" href="#E11:33"><i><font color="Green" title="E30">A8</font></i></a>, <a class="ref" href="ringcat1.html#D10" target="_self" title="RINGCAT1:def.10">Def10</a></i>
;<br/>
<b>hence </b><a NAME="E13:33"/>
<font color="Maroon" title="c7">h</font> <a href="ringcat1.html#K6" title="RINGCAT1:func.6">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">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> <span class="p1">(<span class="default"><font color="Maroon" title="c7">h</font> <a href="ringcat1.html#K6" title="RINGCAT1:func.6">*</a> <font color="Maroon" title="c6">g</font></span>)</span> <a href="ringcat1.html#K6" title="RINGCAT1:func.6">*</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"> verum</span><br/>


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