<?xml version="1.0"?>
<div class="add">

<b>consider </b><font color="Maroon" title="c3">G1'</font>, <font color="Maroon" title="c4">G2'</font>, <font color="Maroon" title="c5">G3'</font> being   <a href="funcsdom.html#NM1" title="FUNCSDOM:NM.1">Ring</a><b> such that </b><br/><a NAME="E2:27_1_1"/><i><font color="Green" title="E17">A2</font></i>: 
( <font color="Maroon" title="c3">G1'</font> <a href="ringcat1.html#R1" title="RINGCAT1:pred.1">&lt;=</a> <font color="Maroon" title="c4">G2'</font> &amp; <font color="Maroon" title="c4">G2'</font> <a href="ringcat1.html#R1" title="RINGCAT1:pred.1">&lt;=</a> <font color="Maroon" title="c5">G3'</font> )
 <b>and </b><br/><a NAME="E3:27_1_1"/><i><font color="Green" title="E18">A3</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="c4">G2'</font>,<font color="Maroon" title="c5">G3'</font>
 <b>and </b><br/><a NAME="E4:27_1_1"/><i><font color="Green" title="E19">A4</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="c3">G1'</font>,<font color="Maroon" title="c4">G2'</font>
 <b>by </b><i><a class="txt" href="ringcat1.html#E1:27"><i><font color="Green" title="E16">A1</font></i></a>, <a class="ref" href="ringcat1.html#T5" target="_self" title="RINGCAT1:th.5">Th5</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c6">g'</font> being   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c4">G2'</font>,<font color="Maroon" title="c5">G3'</font><b> such that </b><br/><a NAME="E6:27_1_1"/><i><font color="Green" title="E20">A5</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> #) <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="c4">G2'</font>,<font color="Maroon" title="c5">G3'</font>,<font color="Maroon" title="c6">g'</font> #)
 <b>and </b><br/><a NAME="E7:27_1_1"/><i><font color="Green" title="E21">A6</font></i>: 
<font color="Maroon" title="c6">g'</font> is <a href="ringcat1.html#V1" title="RINGCAT1:attr.1">linear</a>
 <b>by </b><i><a class="txt" href="ringcat1.html#E2:27_1_1"><i><font color="Green" title="E17">A2</font></i></a>, <a class="txt" href="ringcat1.html#E3:27_1_1"><i><font color="Green" title="E18">A3</font></i></a>, <a class="txt" href="ringcat1.html#E22"><i><font color="Green" title="E14">Lm7</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon" title="c7">f'</font> being   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c3">G1'</font>,<font color="Maroon" title="c4">G2'</font><b> such that </b><br/><a NAME="E9:27_1_1"/><i><font color="Green" title="E22">A7</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> #) <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">G1'</font>,<font color="Maroon" title="c4">G2'</font>,<font color="Maroon" title="c7">f'</font> #)
 <b>and </b><br/><a NAME="E10:27_1_1"/><i><font color="Green" title="E23">A8</font></i>: 
<font color="Maroon" title="c7">f'</font> is <a href="ringcat1.html#V1" title="RINGCAT1:attr.1">linear</a>
 <b>by </b><i><a class="txt" href="ringcat1.html#E2:27_1_1"><i><font color="Green" title="E17">A2</font></i></a>, <a class="txt" href="ringcat1.html#E4:27_1_1"><i><font color="Green" title="E19">A4</font></i></a>, <a class="txt" href="ringcat1.html#E22"><i><font color="Green" title="E14">Lm7</font></i></a></i>;<br/>
<a NAME="E11:27_1_1"/>
<font color="Maroon" title="c6">g'</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c7">f'</font> is <a href="ringcat1.html#V1" title="RINGCAT1:attr.1">linear</a>
 
<b>by </b><i><a class="txt" href="ringcat1.html#E7:27_1_1"><i><font color="Green" title="E21">A6</font></i></a>, <a class="txt" href="ringcat1.html#E10:27_1_1"><i><font color="Green" title="E23">A8</font></i></a>, <a class="ref" href="ringcat1.html#T3" target="_self" title="RINGCAT1:th.3">Th3</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c8">T'</font> =  <a href="ringcat1.html#G1" title="RINGCAT1:aggr.1">RingMorphismStr</a>(# <font color="Maroon" title="c3">G1'</font>,<font color="Maroon" title="c5">G3'</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c6">g'</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c7">f'</font></span>)</span> #) as  <a href="ringcat1.html#V2" title="RINGCAT1:attr.2">strict</a> <a href="ringcat1.html#NM1" title="RINGCAT1:NM.1">RingMorphism</a> <b>by </b><i><a class="txt" href="ringcat1.html#E17"><i><font color="Green" title="E10">Lm5</font></i></a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c8">T'</font>
;<br/>

<b>thus </b><a NAME="E13:27_1_1"/>
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="c8">T'</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>by </b><i><a class="txt" href="ringcat1.html#E6:27_1_1"><i><font color="Green" title="E20">A5</font></i></a>, <a class="txt" href="ringcat1.html#E9:27_1_1"><i><font color="Green" title="E22">A7</font></i></a></i>;<br/>


</div>
