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

<b>set </b><font color="Maroon">S</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> #);<br/>
<a NAME="E1:13"/><i><font color="Green">E31</font></i>: 
 <a href="ringcat1.html#K3" target="_self">fun</a> <font color="Maroon">F</font> is <a href="ringcat1.html#V1" target="_self">linear</a>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E2:13"/>
for <font color="Olive">x</font>, <font color="Olive">y</font> being  <a href="vectsp_1.html#NM3">Scalar</a> of the <a href="ringcat1.html#U1" target="_self">Dom</a> of <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> #) holds  <span class="p1">(<span class="default"><a href="ringcat1.html#K3" target="_self">fun</a> <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> #)</span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Olive">x</font> <a href="rlvect_1.html#K4">+</a> <font color="Olive">y</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ringcat1.html#K3" target="_self">fun</a> <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> #)</span>)</span> <a href="funct_2.html#K3">.</a> <font color="Olive">x</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ringcat1.html#K3" target="_self">fun</a> <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> #)</span>)</span> <a href="funct_2.html#K3">.</a> <font color="Olive">y</font></span>)</span>
 <b>by </b><i/>; <i><font color="Red">:: according to </font></i><a class="ref" href="ringcat1.html#D2" target="_self">RINGCAT1:def 2</a>,<a class="ref" href="ringcat1.html#D6" target="_self">RINGCAT1:def 6</a><br/>

<b>thus </b><a NAME="E3:13"/>
for <font color="Olive">x</font>, <font color="Olive">y</font> being  <a href="vectsp_1.html#NM3">Scalar</a> of the <a href="ringcat1.html#U1" target="_self">Dom</a> of <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> #) holds  <span class="p1">(<span class="default"><a href="ringcat1.html#K3" target="_self">fun</a> <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> #)</span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Olive">x</font> <a href="group_1.html#K1">*</a> <font color="Olive">y</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ringcat1.html#K3" target="_self">fun</a> <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> #)</span>)</span> <a href="funct_2.html#K3">.</a> <font color="Olive">x</font></span>)</span> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ringcat1.html#K3" target="_self">fun</a> <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> #)</span>)</span> <a href="funct_2.html#K3">.</a> <font color="Olive">y</font></span>)</span>
 <b>by </b><i>, </i>;<br/>

<b>thus </b><a NAME="E4:13"/>
<span class="p1">(<span class="default"><a href="ringcat1.html#K3" target="_self">fun</a> <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> #)</span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="group_1.html#K2">1.</a> the <a href="ringcat1.html#U1" target="_self">Dom</a> of <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> #)</span>)</span> <a href="hidden.html#R1">=</a>  <a href="group_1.html#K2">1.</a> the <a href="ringcat1.html#U2" target="_self">Cod</a> of <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> #)
 <b>by </b><i>, </i>;<br/>


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