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













<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:36"/><span class="lab"><font color="Green" title="E23">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="E2:36"/><span class="lab"><font color="Green" title="E24">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>
 <span class="kw">and </span><br/><a NAME="E3:36"/><span class="lab"><font color="Green" title="E25">A3</font></span>: 
<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:()"><span class="comment"><font color="Red">::  thesis: </font></span></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/>

<span class="kw">consider </span><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><span class="kw"> such that </span><br/><a NAME="E5:36"/><span class="lab"><font color="Green" title="E26">A4</font></span>: 
<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> #)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:36"><span class="lab"><font color="Green" title="E23">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="c9">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><span class="kw"> such that </span><br/><a NAME="E7:36"/><span class="lab"><font color="Green" title="E27">A5</font></span>: 
<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="c9">h0</font> #)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:36"><span class="lab"><font color="Green" title="E25">A3</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="c10">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><span class="kw"> such that </span><br/><a NAME="E9:36"/><span class="lab"><font color="Green" title="E28">A6</font></span>: 
<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="c10">g0</font> #)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:36"><span class="lab"><font color="Green" title="E24">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="E10:36"/><span class="lab"><font color="Green" title="E29">A7</font></span>: 
 <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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:36"><span class="lab"><font color="Green" title="E28">A6</font></span></a></span>;<br/>
<a NAME="E11:36"/><span class="lab"><font color="Green" title="E30">A8</font></span>: 
 <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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:36"><span class="lab"><font color="Green" title="E27">A5</font></span></a></span>;<br/>
<a NAME="E12:36"/><span class="kw">then </span><span class="lab"><font color="Green" title="E31">A9</font></span>: 
<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="c9">h0</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c10">g0</font></span>)</span> #)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:36"><span class="lab"><font color="Green" title="E28">A6</font></span></a>, <a class="txt" href="#E7:36"><span class="lab"><font color="Green" title="E27">A5</font></span></a>, <a class="txt" href="#E10:36"><span class="lab"><font color="Green" title="E29">A7</font></span></a>, <a class="ref" href="ringcat1.html#D10" target="_self" title="RINGCAT1:def.10">Def10</a></span>;<br/>
<a NAME="E13:36"/><span class="lab"><font color="Green" title="E32">A10</font></span>: 
 <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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:36"><span class="lab"><font color="Green" title="E28">A6</font></span></a></span>;<br/>
<a NAME="E14:36"/><span class="kw">then </span><span class="lab"><font color="Green" title="E33">A11</font></span>: 
 <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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:36"><span class="lab"><font color="Green" title="E29">A7</font></span></a>, <a class="txt" href="#E11:36"><span class="lab"><font color="Green" title="E30">A8</font></span></a>, <a class="ref" href="ringcat1.html#T11" target="_self" title="RINGCAT1:th.11">Th11</a></span>;<br/>
<a NAME="E15:36"/><span class="lab"><font color="Green" title="E34">A12</font></span>: 
 <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="#E5:36"><span class="lab"><font color="Green" title="E26">A4</font></span></a></span>;<br/>
<a NAME="E16:36"/><span class="kw">then </span><span class="lab"><font color="Green" title="E35">A13</font></span>: 
 <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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:36"><span class="lab"><font color="Green" title="E32">A10</font></span></a>, <a class="txt" href="#E10:36"><span class="lab"><font color="Green" title="E29">A7</font></span></a>, <a class="ref" href="ringcat1.html#T11" target="_self" title="RINGCAT1:th.11">Th11</a></span>;<br/>
<a NAME="E17:36"/>
<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="c10">g0</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c8">f0</font></span>)</span> #)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:36"><span class="lab"><font color="Green" title="E26">A4</font></span></a>, <a class="txt" href="#E9:36"><span class="lab"><font color="Green" title="E28">A6</font></span></a>, <a class="txt" href="#E15:36"><span class="lab"><font color="Green" title="E34">A12</font></span></a>, <a class="txt" href="#E13:36"><span class="lab"><font color="Green" title="E32">A10</font></span></a>, <a class="ref" href="ringcat1.html#D10" target="_self" title="RINGCAT1:def.10">Def10</a></span>;<br/>
<span class="kw">then </span><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="c9">h0</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">g0</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c8">f0</font></span>)</span></span>)</span> #)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:36"><span class="lab"><font color="Green" title="E27">A5</font></span></a>, <a class="txt" href="#E11:36"><span class="lab"><font color="Green" title="E30">A8</font></span></a>, <a class="txt" href="#E16:36"><span class="lab"><font color="Green" title="E35">A13</font></span></a>, <a class="ref" href="ringcat1.html#D10" target="_self" title="RINGCAT1:def.10">Def10</a></span>
<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="c9">h0</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c10">g0</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c8">f0</font></span>)</span> #)
<span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#T55" title="RELAT_1:th.55">RELAT_1:55</a></span>
<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>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:36"><span class="lab"><font color="Green" title="E26">A4</font></span></a>, <a class="txt" href="#E15:36"><span class="lab"><font color="Green" title="E34">A12</font></span></a>, <a class="txt" href="#E12:36"><span class="lab"><font color="Green" title="E31">A9</font></span></a>, <a class="txt" href="#E14:36"><span class="lab"><font color="Green" title="E33">A11</font></span></a>, <a class="ref" href="ringcat1.html#D10" target="_self" title="RINGCAT1:def.10">Def10</a></span>
;<br/>
<span class="kw">hence </span><a NAME="E19:36"/>
<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:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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