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<b>let </b><font color="Maroon">S1</font> be  <a href="ringcat1.html#V2" target="_self">strict</a> <a href="ringcat1.html#NM1">RingMorphism</a>, <font color="Maroon">S2</font> be  <a href="ringcat1.html#V2" target="_self">strict</a> <a href="ringcat1.html#NM1">RingMorphism</a>;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:26_1_2"/><i><font color="Green">E69</font></i>: 
for <font color="Olive">G1</font>, <font color="Olive">G2</font>, <font color="Olive">G3</font> being  <a href="funcsdom.html#NM1">Ring</a><br/> for <font color="Olive">g</font> being  <a href="struct_0.html#NM6">Function</a> of <font color="Olive">G2</font>,<font color="Olive">G3</font><br/> for <font color="Olive">f</font> being  <a href="struct_0.html#NM6">Function</a> of <font color="Olive">G1</font>,<font color="Olive">G2</font>  st  <a href="ringcat1.html#G1" target="_self">RingMorphismStr</a>(# the <a href="ringcat1.html#U1" target="_self">Dom</a> of <font color="Maroon">G</font>,the <a href="ringcat1.html#U2" target="_self">Cod</a> of <font color="Maroon">G</font>,the <a href="ringcat1.html#U3" target="_self">Fun</a> of <font color="Maroon">G</font> #) <a href="hidden.html#R1">=</a>  <a href="ringcat1.html#G1" target="_self">RingMorphismStr</a>(# <font color="Olive">G2</font>,<font color="Olive">G3</font>,<font color="Olive">g</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">F</font>,the <a href="ringcat1.html#U2" target="_self">Cod</a> of <font color="Maroon">F</font>,the <a href="ringcat1.html#U3" target="_self">Fun</a> of <font color="Maroon">F</font> #) <a href="hidden.html#R1">=</a>  <a href="ringcat1.html#G1" target="_self">RingMorphismStr</a>(# <font color="Olive">G1</font>,<font color="Olive">G2</font>,<font color="Olive">f</font> #) holds <br/><font color="Maroon">S1</font> <a href="hidden.html#R1">=</a>  <a href="ringcat1.html#G1" target="_self">RingMorphismStr</a>(# <font color="Olive">G1</font>,<font color="Olive">G3</font>,<span class="p1">(<span class="default"><font color="Olive">g</font> <a href="partfun1.html#K1">*</a> <font color="Olive">f</font></span>)</span> #)
 <b>and </b><br/><a NAME="E2:26_1_2"/><i><font color="Green">E70</font></i>: 
for <font color="Olive">G1</font>, <font color="Olive">G2</font>, <font color="Olive">G3</font> being  <a href="funcsdom.html#NM1">Ring</a><br/> for <font color="Olive">g</font> being  <a href="struct_0.html#NM6">Function</a> of <font color="Olive">G2</font>,<font color="Olive">G3</font><br/> for <font color="Olive">f</font> being  <a href="struct_0.html#NM6">Function</a> of <font color="Olive">G1</font>,<font color="Olive">G2</font>  st  <a href="ringcat1.html#G1" target="_self">RingMorphismStr</a>(# the <a href="ringcat1.html#U1" target="_self">Dom</a> of <font color="Maroon">G</font>,the <a href="ringcat1.html#U2" target="_self">Cod</a> of <font color="Maroon">G</font>,the <a href="ringcat1.html#U3" target="_self">Fun</a> of <font color="Maroon">G</font> #) <a href="hidden.html#R1">=</a>  <a href="ringcat1.html#G1" target="_self">RingMorphismStr</a>(# <font color="Olive">G2</font>,<font color="Olive">G3</font>,<font color="Olive">g</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">F</font>,the <a href="ringcat1.html#U2" target="_self">Cod</a> of <font color="Maroon">F</font>,the <a href="ringcat1.html#U3" target="_self">Fun</a> of <font color="Maroon">F</font> #) <a href="hidden.html#R1">=</a>  <a href="ringcat1.html#G1" target="_self">RingMorphismStr</a>(# <font color="Olive">G1</font>,<font color="Olive">G2</font>,<font color="Olive">f</font> #) holds <br/><font color="Maroon">S2</font> <a href="hidden.html#R1">=</a>  <a href="ringcat1.html#G1" target="_self">RingMorphismStr</a>(# <font color="Olive">G1</font>,<font color="Olive">G3</font>,<span class="p1">(<span class="default"><font color="Olive">g</font> <a href="partfun1.html#K1">*</a> <font color="Olive">f</font></span>)</span> #)
 ;<br/>

<b>consider </b><font color="Maroon">G2'</font> being   <a href="funcsdom.html#NM1">Ring</a>, <font color="Maroon">G3'</font> being   <a href="funcsdom.html#NM1">Ring</a><b> such that </b><br/><a NAME="E4:26_1_2"/><i><font color="Green">E71</font></i>: 
<font color="Maroon">G2'</font> <a href="ringcat1.html#R1" target="_self">&lt;=</a> <font color="Maroon">G3'</font>
 <b>and </b><br/><a NAME="E5:26_1_2"/><i><font color="Green">E72</font></i>: 
(  <a href="ringcat1.html#K1" target="_self">dom</a> <font color="Maroon">G</font> <a href="hidden.html#R1">=</a> <font color="Maroon">G2'</font> &amp;  <a href="ringcat1.html#K2" target="_self">cod</a> <font color="Maroon">G</font> <a href="hidden.html#R1">=</a> <font color="Maroon">G3'</font> )
 <b>and </b><br/><a NAME="E6:26_1_2"/><i><font color="Green">E73</font></i>: 
 <a href="ringcat1.html#G1" target="_self">RingMorphismStr</a>(# the <a href="ringcat1.html#U1" target="_self">Dom</a> of <font color="Maroon">G</font>,the <a href="ringcat1.html#U2" target="_self">Cod</a> of <font color="Maroon">G</font>,the <a href="ringcat1.html#U3" target="_self">Fun</a> of <font color="Maroon">G</font> #) is    <a href="ringcat1.html#M1" target="_self">Morphism</a> of <font color="Maroon">G2'</font>,<font color="Maroon">G3'</font>
 <b>by </b><i/>;<br/>
<b>reconsider </b><font color="Maroon">G'</font> =  <a href="ringcat1.html#G1" target="_self">RingMorphismStr</a>(# the <a href="ringcat1.html#U1" target="_self">Dom</a> of <font color="Maroon">G</font>,the <a href="ringcat1.html#U2" target="_self">Cod</a> of <font color="Maroon">G</font>,the <a href="ringcat1.html#U3" target="_self">Fun</a> of <font color="Maroon">G</font> #) as    <a href="ringcat1.html#M1" target="_self">Morphism</a> of <font color="Maroon">G2'</font>,<font color="Maroon">G3'</font> <b>by </b><i/>;<br/>
<b>consider </b><font color="Maroon">G1'</font> being   <a href="funcsdom.html#NM1">Ring</a>, <font color="Maroon">G2''</font> being   <a href="funcsdom.html#NM1">Ring</a><b> such that </b><br/><a NAME="E9:26_1_2"/>
<font color="Maroon">G1'</font> <a href="ringcat1.html#R1" target="_self">&lt;=</a> <font color="Maroon">G2''</font>
 <b>and </b><br/><a NAME="E10:26_1_2"/><i><font color="Green">E76</font></i>: 
(  <a href="ringcat1.html#K1" target="_self">dom</a> <font color="Maroon">F</font> <a href="hidden.html#R1">=</a> <font color="Maroon">G1'</font> &amp;  <a href="ringcat1.html#K2" target="_self">cod</a> <font color="Maroon">F</font> <a href="hidden.html#R1">=</a> <font color="Maroon">G2''</font> )
 <b>and </b><br/><a NAME="E11:26_1_2"/><i><font color="Green">E77</font></i>: 
 <a href="ringcat1.html#G1" target="_self">RingMorphismStr</a>(# the <a href="ringcat1.html#U1" target="_self">Dom</a> of <font color="Maroon">F</font>,the <a href="ringcat1.html#U2" target="_self">Cod</a> of <font color="Maroon">F</font>,the <a href="ringcat1.html#U3" target="_self">Fun</a> of <font color="Maroon">F</font> #) is    <a href="ringcat1.html#M1" target="_self">Morphism</a> of <font color="Maroon">G1'</font>,<font color="Maroon">G2''</font>
 <b>by </b><i/>;<br/>
<b>reconsider </b><font color="Maroon">F'</font> =  <a href="ringcat1.html#G1" target="_self">RingMorphismStr</a>(# the <a href="ringcat1.html#U1" target="_self">Dom</a> of <font color="Maroon">F</font>,the <a href="ringcat1.html#U2" target="_self">Cod</a> of <font color="Maroon">F</font>,the <a href="ringcat1.html#U3" target="_self">Fun</a> of <font color="Maroon">F</font> #) as    <a href="ringcat1.html#M1" target="_self">Morphism</a> of <font color="Maroon">G1'</font>,<font color="Maroon">G2''</font> <b>by </b><i/>;<br/>
<b>reconsider </b><font color="Maroon">F'</font> = <font color="Maroon">F'</font> as    <a href="ringcat1.html#M1" target="_self">Morphism</a> of <font color="Maroon">G1'</font>,<font color="Maroon">G2'</font> <b>by </b><i>, , </i>;<br/>
<b>consider </b><font color="Maroon">g'</font> being   <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">G2'</font>,<font color="Maroon">G3'</font><b> such that </b><br/><a NAME="E15:26_1_2"/><i><font color="Green">E78</font></i>: 
<font color="Maroon">G'</font> <a href="hidden.html#R1">=</a>  <a href="ringcat1.html#G1" target="_self">RingMorphismStr</a>(# <font color="Maroon">G2'</font>,<font color="Maroon">G3'</font>,<font color="Maroon">g'</font> #)
 <b>by </b><i>, </i>;<br/>
<b>consider </b><font color="Maroon">f'</font> being   <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">G1'</font>,<font color="Maroon">G2'</font><b> such that </b><br/><a NAME="E17:26_1_2"/><i><font color="Green">E79</font></i>: 
<font color="Maroon">F'</font> <a href="hidden.html#R1">=</a>  <a href="ringcat1.html#G1" target="_self">RingMorphismStr</a>(# <font color="Maroon">G1'</font>,<font color="Maroon">G2'</font>,<font color="Maroon">f'</font> #)
 <b>by </b><i>, , </i>;<br/>
<b>thus </b><font color="Maroon">S1</font> = 
 <a href="ringcat1.html#G1" target="_self">RingMorphismStr</a>(# <font color="Maroon">G1'</font>,<font color="Maroon">G3'</font>,<span class="p1">(<span class="default"><font color="Maroon">g'</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">f'</font></span>)</span> #)
<b>by </b><i>, <a class="ref" href="ringcat1.html#D2" target="_self">Def2</a>, <a class="ref" href="ringcat1.html#T3" target="_self">Th3</a></i>
<br/>.= 
<font color="Maroon">S2</font>
<b>by </b><i>, <a class="ref" href="ringcat1.html#D2" target="_self">Def2</a>, <a class="ref" href="ringcat1.html#T3" target="_self">Th3</a></i>
;<br/>


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