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<b>let </b><font color="Maroon" title="c3">S1</font>, <font color="Maroon" title="c4">S2</font> be  <a href="ringcat1.html#V2" title="RINGCAT1:attr.2">strict</a> <a href="ringcat1.html#NM1" title="RINGCAT1:NM.1">RingMorphism</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">G1</font>, <font color="Olive" title="b2">G2</font>, <font color="Olive" title="b3">G3</font> being  <a href="funcsdom.html#NM1" title="FUNCSDOM:NM.1">Ring</a><br/> for <font color="Olive" title="b4">g</font> being  <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Olive" title="b2">G2</font>,<font color="Olive" title="b3">G3</font><br/> for <font color="Olive" title="b5">f</font> being  <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Olive" title="b1">G1</font>,<font color="Olive" title="b2">G2</font>  st  <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="c1">G</font>,the <a href="ringcat1.html#U2" title="RINGCAT1:sel.2">Cod</a> of <font color="Maroon" title="c1">G</font>,the <a href="ringcat1.html#U3" title="RINGCAT1:sel.3">Fun</a> of <font color="Maroon" title="c1">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="Olive" title="b2">G2</font>,<font color="Olive" title="b3">G3</font>,<font color="Olive" title="b4">g</font> #) &amp;  <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="c2">F</font>,the <a href="ringcat1.html#U2" title="RINGCAT1:sel.2">Cod</a> of <font color="Maroon" title="c2">F</font>,the <a href="ringcat1.html#U3" title="RINGCAT1:sel.3">Fun</a> of <font color="Maroon" title="c2">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="Olive" title="b1">G1</font>,<font color="Olive" title="b2">G2</font>,<font color="Olive" title="b5">f</font> #) holds <br/><font color="Maroon" title="c3">S1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ringcat1.html#G1" title="RINGCAT1:aggr.1">RingMorphismStr</a>(# <font color="Olive" title="b1">G1</font>,<font color="Olive" title="b3">G3</font>,<span class="p1">(<span class="default"><font color="Olive" title="b4">g</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Olive" title="b5">f</font></span>)</span> #) ) &amp; ( for <font color="Olive" title="b1">G1</font>, <font color="Olive" title="b2">G2</font>, <font color="Olive" title="b3">G3</font> being  <a href="funcsdom.html#NM1" title="FUNCSDOM:NM.1">Ring</a><br/> for <font color="Olive" title="b4">g</font> being  <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Olive" title="b2">G2</font>,<font color="Olive" title="b3">G3</font><br/> for <font color="Olive" title="b5">f</font> being  <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Olive" title="b1">G1</font>,<font color="Olive" title="b2">G2</font>  st  <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="c1">G</font>,the <a href="ringcat1.html#U2" title="RINGCAT1:sel.2">Cod</a> of <font color="Maroon" title="c1">G</font>,the <a href="ringcat1.html#U3" title="RINGCAT1:sel.3">Fun</a> of <font color="Maroon" title="c1">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="Olive" title="b2">G2</font>,<font color="Olive" title="b3">G3</font>,<font color="Olive" title="b4">g</font> #) &amp;  <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="c2">F</font>,the <a href="ringcat1.html#U2" title="RINGCAT1:sel.2">Cod</a> of <font color="Maroon" title="c2">F</font>,the <a href="ringcat1.html#U3" title="RINGCAT1:sel.3">Fun</a> of <font color="Maroon" title="c2">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="Olive" title="b1">G1</font>,<font color="Olive" title="b2">G2</font>,<font color="Olive" title="b5">f</font> #) holds <br/><font color="Maroon" title="c4">S2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ringcat1.html#G1" title="RINGCAT1:aggr.1">RingMorphismStr</a>(# <font color="Olive" title="b1">G1</font>,<font color="Olive" title="b3">G3</font>,<span class="p1">(<span class="default"><font color="Olive" title="b4">g</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Olive" title="b5">f</font></span>)</span> #) ) implies <font color="Maroon" title="c3">S1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">S2</font> )</span><br/>

<b>assume </b><b>that </b><br/><a NAME="E1:27_1_2"/><i><font color="Green" title="E17">A9</font></i>: 
for <font color="Olive" title="b1">G1</font>, <font color="Olive" title="b2">G2</font>, <font color="Olive" title="b3">G3</font> being  <a href="funcsdom.html#NM1" title="FUNCSDOM:NM.1">Ring</a><br/> for <font color="Olive" title="b4">g</font> being  <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Olive" title="b2">G2</font>,<font color="Olive" title="b3">G3</font><br/> for <font color="Olive" title="b5">f</font> being  <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Olive" title="b1">G1</font>,<font color="Olive" title="b2">G2</font>  st  <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="c1">G</font>,the <a href="ringcat1.html#U2" title="RINGCAT1:sel.2">Cod</a> of <font color="Maroon" title="c1">G</font>,the <a href="ringcat1.html#U3" title="RINGCAT1:sel.3">Fun</a> of <font color="Maroon" title="c1">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="Olive" title="b2">G2</font>,<font color="Olive" title="b3">G3</font>,<font color="Olive" title="b4">g</font> #) &amp;  <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="c2">F</font>,the <a href="ringcat1.html#U2" title="RINGCAT1:sel.2">Cod</a> of <font color="Maroon" title="c2">F</font>,the <a href="ringcat1.html#U3" title="RINGCAT1:sel.3">Fun</a> of <font color="Maroon" title="c2">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="Olive" title="b1">G1</font>,<font color="Olive" title="b2">G2</font>,<font color="Olive" title="b5">f</font> #) holds <br/><font color="Maroon" title="c3">S1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ringcat1.html#G1" title="RINGCAT1:aggr.1">RingMorphismStr</a>(# <font color="Olive" title="b1">G1</font>,<font color="Olive" title="b3">G3</font>,<span class="p1">(<span class="default"><font color="Olive" title="b4">g</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Olive" title="b5">f</font></span>)</span> #)
 <b>and </b><br/><a NAME="E2:27_1_2"/><i><font color="Green" title="E18">A10</font></i>: 
for <font color="Olive" title="b1">G1</font>, <font color="Olive" title="b2">G2</font>, <font color="Olive" title="b3">G3</font> being  <a href="funcsdom.html#NM1" title="FUNCSDOM:NM.1">Ring</a><br/> for <font color="Olive" title="b4">g</font> being  <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Olive" title="b2">G2</font>,<font color="Olive" title="b3">G3</font><br/> for <font color="Olive" title="b5">f</font> being  <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Olive" title="b1">G1</font>,<font color="Olive" title="b2">G2</font>  st  <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="c1">G</font>,the <a href="ringcat1.html#U2" title="RINGCAT1:sel.2">Cod</a> of <font color="Maroon" title="c1">G</font>,the <a href="ringcat1.html#U3" title="RINGCAT1:sel.3">Fun</a> of <font color="Maroon" title="c1">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="Olive" title="b2">G2</font>,<font color="Olive" title="b3">G3</font>,<font color="Olive" title="b4">g</font> #) &amp;  <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="c2">F</font>,the <a href="ringcat1.html#U2" title="RINGCAT1:sel.2">Cod</a> of <font color="Maroon" title="c2">F</font>,the <a href="ringcat1.html#U3" title="RINGCAT1:sel.3">Fun</a> of <font color="Maroon" title="c2">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="Olive" title="b1">G1</font>,<font color="Olive" title="b2">G2</font>,<font color="Olive" title="b5">f</font> #) holds <br/><font color="Maroon" title="c4">S2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ringcat1.html#G1" title="RINGCAT1:aggr.1">RingMorphismStr</a>(# <font color="Olive" title="b1">G1</font>,<font color="Olive" title="b3">G3</font>,<span class="p1">(<span class="default"><font color="Olive" title="b4">g</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Olive" title="b5">f</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="c3">S1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">S2</font></span><br/>

<b>consider </b><font color="Maroon" title="c5">G2'</font>, <font color="Maroon" title="c6">G3'</font> being   <a href="funcsdom.html#NM1" title="FUNCSDOM:NM.1">Ring</a><b> such that </b><br/><a NAME="E4:27_1_2"/><i><font color="Green" title="E19">A11</font></i>: 
<font color="Maroon" title="c5">G2'</font> <a href="ringcat1.html#R1" title="RINGCAT1:pred.1">&lt;=</a> <font color="Maroon" title="c6">G3'</font>
 <b>and </b><br/><a NAME="E5:27_1_2"/><i><font color="Green" title="E20">A12</font></i>: 
(  <a href="ringcat1.html#K1" title="RINGCAT1:func.1">dom</a> <font color="Maroon" title="c1">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">G2'</font> &amp;  <a href="ringcat1.html#K2" title="RINGCAT1:func.2">cod</a> <font color="Maroon" title="c1">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">G3'</font> )
 <b>and </b><br/><a NAME="E6:27_1_2"/><i><font color="Green" title="E21">A13</font></i>: 
 <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="c1">G</font>,the <a href="ringcat1.html#U2" title="RINGCAT1:sel.2">Cod</a> of <font color="Maroon" title="c1">G</font>,the <a href="ringcat1.html#U3" title="RINGCAT1:sel.3">Fun</a> of <font color="Maroon" title="c1">G</font> #) is    <a href="ringcat1.html#M1" title="RINGCAT1:mode.1">Morphism</a> of <font color="Maroon" title="c5">G2'</font>,<font color="Maroon" title="c6">G3'</font>
 <b>by </b><i><a class="txt" href="ringcat1.html#E18"><i><font color="Green" title="E11">Lm6</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c7">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="c1">G</font>,the <a href="ringcat1.html#U2" title="RINGCAT1:sel.2">Cod</a> of <font color="Maroon" title="c1">G</font>,the <a href="ringcat1.html#U3" title="RINGCAT1:sel.3">Fun</a> of <font color="Maroon" title="c1">G</font> #) as    <a href="ringcat1.html#M1" title="RINGCAT1:mode.1">Morphism</a> of <font color="Maroon" title="c5">G2'</font>,<font color="Maroon" title="c6">G3'</font> <b>by </b><i><a class="txt" href="#E6:27_1_2"><i><font color="Green" title="E21">A13</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon" title="c8">G1'</font>, <font color="Maroon" title="c9">G2''</font> being   <a href="funcsdom.html#NM1" title="FUNCSDOM:NM.1">Ring</a><b> such that </b><br/><a NAME="E9:27_1_2"/>
<font color="Maroon" title="c8">G1'</font> <a href="ringcat1.html#R1" title="RINGCAT1:pred.1">&lt;=</a> <font color="Maroon" title="c9">G2''</font>
 <b>and </b><br/><a NAME="E10:27_1_2"/><i><font color="Green" title="E22">A14</font></i>: 
(  <a href="ringcat1.html#K1" title="RINGCAT1:func.1">dom</a> <font color="Maroon" title="c2">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">G1'</font> &amp;  <a href="ringcat1.html#K2" title="RINGCAT1:func.2">cod</a> <font color="Maroon" title="c2">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">G2''</font> )
 <b>and </b><br/><a NAME="E11:27_1_2"/><i><font color="Green" title="E23">A15</font></i>: 
 <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="c2">F</font>,the <a href="ringcat1.html#U2" title="RINGCAT1:sel.2">Cod</a> of <font color="Maroon" title="c2">F</font>,the <a href="ringcat1.html#U3" title="RINGCAT1:sel.3">Fun</a> of <font color="Maroon" title="c2">F</font> #) is    <a href="ringcat1.html#M1" title="RINGCAT1:mode.1">Morphism</a> of <font color="Maroon" title="c8">G1'</font>,<font color="Maroon" title="c9">G2''</font>
 <b>by </b><i><a class="txt" href="ringcat1.html#E18"><i><font color="Green" title="E11">Lm6</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c10">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="c2">F</font>,the <a href="ringcat1.html#U2" title="RINGCAT1:sel.2">Cod</a> of <font color="Maroon" title="c2">F</font>,the <a href="ringcat1.html#U3" title="RINGCAT1:sel.3">Fun</a> of <font color="Maroon" title="c2">F</font> #) as    <a href="ringcat1.html#M1" title="RINGCAT1:mode.1">Morphism</a> of <font color="Maroon" title="c8">G1'</font>,<font color="Maroon" title="c9">G2''</font> <b>by </b><i><a class="txt" href="#E11:27_1_2"><i><font color="Green" title="E23">A15</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c11">F'</font> = <font color="Maroon" title="c10">F'</font> as    <a href="ringcat1.html#M1" title="RINGCAT1:mode.1">Morphism</a> of <font color="Maroon" title="c8">G1'</font>,<font color="Maroon" title="c5">G2'</font> <b>by </b><i><a class="txt" href="#E1:27"><i><font color="Green" title="E16">A1</font></i></a>, <a class="txt" href="#E5:27_1_2"><i><font color="Green" title="E20">A12</font></i></a>, <a class="txt" href="#E10:27_1_2"><i><font color="Green" title="E22">A14</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon" title="c12">g'</font> being   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c5">G2'</font>,<font color="Maroon" title="c6">G3'</font><b> such that </b><br/><a NAME="E15:27_1_2"/><i><font color="Green" title="E24">A16</font></i>: 
<font color="Maroon" title="c7">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="c5">G2'</font>,<font color="Maroon" title="c6">G3'</font>,<font color="Maroon" title="c12">g'</font> #)
 <b>by </b><i><a class="txt" href="#E4:27_1_2"><i><font color="Green" title="E19">A11</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="c13">f'</font> being   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c8">G1'</font>,<font color="Maroon" title="c5">G2'</font><b> such that </b><br/><a NAME="E17:27_1_2"/><i><font color="Green" title="E25">A17</font></i>: 
<font color="Maroon" title="c11">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="c8">G1'</font>,<font color="Maroon" title="c5">G2'</font>,<font color="Maroon" title="c13">f'</font> #)
 <b>by </b><i><a class="txt" href="#E1:27"><i><font color="Green" title="E16">A1</font></i></a>, <a class="txt" href="#E5:27_1_2"><i><font color="Green" title="E20">A12</font></i></a>, <a class="txt" href="#E10:27_1_2"><i><font color="Green" title="E22">A14</font></i></a></i>;<br/>
<b>thus </b><font color="Maroon" title="c3">S1</font> = 
 <a href="ringcat1.html#G1" title="RINGCAT1:aggr.1">RingMorphismStr</a>(# <font color="Maroon" title="c8">G1'</font>,<font color="Maroon" title="c6">G3'</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c12">g'</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c13">f'</font></span>)</span> #)
<b>by </b><i><a class="txt" href="#E1:27_1_2"><i><font color="Green" title="E17">A9</font></i></a>, <a class="txt" href="#E15:27_1_2"><i><font color="Green" title="E24">A16</font></i></a>, <a class="txt" href="#E17:27_1_2"><i><font color="Green" title="E25">A17</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c4">S2</font>
<b>by </b><i><a class="txt" href="#E2:27_1_2"><i><font color="Green" title="E18">A10</font></i></a>, <a class="txt" href="#E15:27_1_2"><i><font color="Green" title="E24">A16</font></i></a>, <a class="txt" href="#E17:27_1_2"><i><font color="Green" title="E25">A17</font></i></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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