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<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[  <a href="ringcat1.html#NM1">RingMorphism</a>,  <a href="ringcat1.html#NM1">RingMorphism</a>] means  <a href="ringcat1.html#K1" target="_self">dom</a> <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="ringcat1.html#K2" target="_self">cod</a> <font color="Maroon">a<sub>2</sub></font>;<br/>
<b>let </b><font color="Maroon">c<sub>1</sub></font>, <font color="Maroon">c<sub>2</sub></font> be   <a href="ringcat1.html#NM1">RingMorphism</a>;<br/>

<b>assume </b><a NAME="E1:21"/><i><font color="Green">E13</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>]
 ;<br/>

<a NAME="E2:21"/><i><font color="Green">E14</font></i>: 
ex <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being  <a href="funcsdom.html#NM1">Ring</a> st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="ringcat1.html#R1" target="_self">&lt;=</a> <font color="Olive">b<sub>2</sub></font> &amp;  <a href="ringcat1.html#K1" target="_self">dom</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> &amp;  <a href="ringcat1.html#K2" target="_self">cod</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>2</sub></font> &amp;  <a href="ringcat1.html#G1" target="_self">RingMorphismStr</a>(# the <a href="ringcat1.html#U1" target="_self">Dom</a> of <font color="Maroon">c<sub>1</sub></font>,the <a href="ringcat1.html#U2" target="_self">Cod</a> of <font color="Maroon">c<sub>1</sub></font>,the <a href="ringcat1.html#U3" target="_self">Fun</a> of <font color="Maroon">c<sub>1</sub></font> #) is    <a href="ringcat1.html#M1" target="_self">Morphism</a> of <font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font> )
 
<b>by </b><i><a class="txt" href="ringcat1.html#E18"><i><font color="Green">Lemma11</font></i></a></i>;<br/>
<a NAME="E3:21"/>
ex <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being  <a href="funcsdom.html#NM1">Ring</a> st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="ringcat1.html#R1" target="_self">&lt;=</a> <font color="Olive">b<sub>2</sub></font> &amp;  <a href="ringcat1.html#K1" target="_self">dom</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> &amp;  <a href="ringcat1.html#K2" target="_self">cod</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>2</sub></font> &amp;  <a href="ringcat1.html#G1" target="_self">RingMorphismStr</a>(# the <a href="ringcat1.html#U1" target="_self">Dom</a> of <font color="Maroon">c<sub>2</sub></font>,the <a href="ringcat1.html#U2" target="_self">Cod</a> of <font color="Maroon">c<sub>2</sub></font>,the <a href="ringcat1.html#U3" target="_self">Fun</a> of <font color="Maroon">c<sub>2</sub></font> #) is    <a href="ringcat1.html#M1" target="_self">Morphism</a> of <font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font> )
 
<b>by </b><i><a class="txt" href="ringcat1.html#E18"><i><font color="Green">Lemma11</font></i></a></i>;<br/>
<b>hence </b><a NAME="E4:21"/>
ex <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font>, <font color="Olive">b<sub>3</sub></font> being  <a href="funcsdom.html#NM1">Ring</a> st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="ringcat1.html#R1" target="_self">&lt;=</a> <font color="Olive">b<sub>2</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="ringcat1.html#R1" target="_self">&lt;=</a> <font color="Olive">b<sub>3</sub></font> &amp;  <a href="ringcat1.html#G1" target="_self">RingMorphismStr</a>(# the <a href="ringcat1.html#U1" target="_self">Dom</a> of <font color="Maroon">c<sub>1</sub></font>,the <a href="ringcat1.html#U2" target="_self">Cod</a> of <font color="Maroon">c<sub>1</sub></font>,the <a href="ringcat1.html#U3" target="_self">Fun</a> of <font color="Maroon">c<sub>1</sub></font> #) is    <a href="ringcat1.html#M1" target="_self">Morphism</a> of <font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>3</sub></font> &amp;  <a href="ringcat1.html#G1" target="_self">RingMorphismStr</a>(# the <a href="ringcat1.html#U1" target="_self">Dom</a> of <font color="Maroon">c<sub>2</sub></font>,the <a href="ringcat1.html#U2" target="_self">Cod</a> of <font color="Maroon">c<sub>2</sub></font>,the <a href="ringcat1.html#U3" target="_self">Fun</a> of <font color="Maroon">c<sub>2</sub></font> #) is    <a href="ringcat1.html#M1" target="_self">Morphism</a> of <font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font> )
 <b>by </b><i><a class="txt" href="ringcat1.html#E1:21"><i><font color="Green">E13</font></i></a>, <a class="txt" href="ringcat1.html#E2:21"><i><font color="Green">E14</font></i></a></i>;<br/>


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