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<b>let </b><font color="Maroon">c<sub>1</sub></font> be   <a href="ringcat1.html#NM1">RingMorphism</a>;<br/>

<b>set </b><font color="Maroon">c<sub>2</sub></font> = the <a href="ringcat1.html#U1" target="_self">Dom</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>
<b>set </b><font color="Maroon">c<sub>3</sub></font> = the <a href="ringcat1.html#U2" target="_self">Cod</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>
<a NAME="E1:20"/>
(  <a href="ringcat1.html#K1" target="_self">dom</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> the <a href="ringcat1.html#U1" target="_self">Dom</a> of <font color="Maroon">c<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> the <a href="ringcat1.html#U2" target="_self">Cod</a> of <font color="Maroon">c<sub>1</sub></font> )
 
;<br/>
<a NAME="E2:20"/><b>then </b><i><font color="Green">E12</font></i>: 
the <a href="ringcat1.html#U1" target="_self">Dom</a> of <font color="Maroon">c<sub>1</sub></font> <a href="ringcat1.html#R1" target="_self">&lt;=</a> the <a href="ringcat1.html#U2" target="_self">Cod</a> of <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="ringcat1.html#D8" target="_self">Def8</a></i>;<br/>
<b>take </b>
the <a href="ringcat1.html#U1" target="_self">Dom</a> of <font color="Maroon">c<sub>1</sub></font>
;<br/>

<b>take </b>
the <a href="ringcat1.html#U2" target="_self">Cod</a> of <font color="Maroon">c<sub>1</sub></font>
;<br/>

<b>reconsider </b><font color="Maroon">c<sub>4</sub></font> =  <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> #) as   <a href="ringcat1.html#NM1">RingMorphism</a> <b>by </b><i><a class="txt" href="ringcat1.html#E13"><i><font color="Green">Lemma6</font></i></a></i>;<br/>
<a NAME="E4:20"/>
(  <a href="ringcat1.html#K1" target="_self">dom</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> the <a href="ringcat1.html#U1" target="_self">Dom</a> of <font color="Maroon">c<sub>1</sub></font> &amp;  <a href="ringcat1.html#K2" target="_self">cod</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> the <a href="ringcat1.html#U2" target="_self">Cod</a> of <font color="Maroon">c<sub>1</sub></font> )
 
;<br/>
<b>hence </b><a NAME="E5:20"/>
( the <a href="ringcat1.html#U1" target="_self">Dom</a> of <font color="Maroon">c<sub>1</sub></font> <a href="ringcat1.html#R1" target="_self">&lt;=</a> the <a href="ringcat1.html#U2" target="_self">Cod</a> of <font color="Maroon">c<sub>1</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> the <a href="ringcat1.html#U1" target="_self">Dom</a> of <font color="Maroon">c<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> the <a href="ringcat1.html#U2" target="_self">Cod</a> of <font color="Maroon">c<sub>1</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 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> )
 <b>by </b><i><a class="txt" href="ringcat1.html#E2:20"><i><font color="Green">E12</font></i></a>, <a class="ref" href="ringcat1.html#D9" target="_self">Def9</a></i>;<br/>


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