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<span class="kw">let </span><font color="Maroon" title="c1">F</font> be   <a href="ringcat1.html#NM1" title="RINGCAT1:NM.1">RingMorphism</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <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">F</font>,the <a href="ringcat1.html#U2" title="RINGCAT1:sel.2">Cod</a> of <font color="Maroon" title="c1">F</font>,the <a href="ringcat1.html#U3" title="RINGCAT1:sel.3">Fun</a> of <font color="Maroon" title="c1">F</font> #) is  <a href="ringcat1.html#V3" title="RINGCAT1:attr.3">RingMorphism-like</a> </span><br/>

<span class="kw">set </span><font color="Maroon" title="c2">S</font> =  <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">F</font>,the <a href="ringcat1.html#U2" title="RINGCAT1:sel.2">Cod</a> of <font color="Maroon" title="c1">F</font>,the <a href="ringcat1.html#U3" title="RINGCAT1:sel.3">Fun</a> of <font color="Maroon" title="c1">F</font> #);<br/>
<a NAME="E1:16"/><span class="lab"><font color="Green" title="E7">A1</font></span>: 
 <a href="ringcat1.html#K3" title="RINGCAT1:func.3">fun</a> <font color="Maroon" title="c1">F</font> is  <a href="ringcat1.html#V1" title="RINGCAT1:attr.1">linear</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="ringcat1.html#D6" target="_self" title="RINGCAT1:def.6">Def6</a></span>;<br/>
<span class="kw">hence </span><a NAME="E2:16"/>
for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font> being  <a href="vectsp_1.html#NM4" title="VECTSP_1:NM.4">Scalar</a> of the <a href="ringcat1.html#U1" title="RINGCAT1:sel.1">Dom</a> of <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">F</font>,the <a href="ringcat1.html#U2" title="RINGCAT1:sel.2">Cod</a> of <font color="Maroon" title="c1">F</font>,the <a href="ringcat1.html#U3" title="RINGCAT1:sel.3">Fun</a> of <font color="Maroon" title="c1">F</font> #) holds  <span class="p1">(<span class="default"><a href="ringcat1.html#K3" title="RINGCAT1:func.3">fun</a> <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">F</font>,the <a href="ringcat1.html#U2" title="RINGCAT1:sel.2">Cod</a> of <font color="Maroon" title="c1">F</font>,the <a href="ringcat1.html#U3" title="RINGCAT1:sel.3">Fun</a> of <font color="Maroon" title="c1">F</font> #)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">x</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Olive" title="b2">y</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ringcat1.html#K3" title="RINGCAT1:func.3">fun</a> <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">F</font>,the <a href="ringcat1.html#U2" title="RINGCAT1:sel.2">Cod</a> of <font color="Maroon" title="c1">F</font>,the <a href="ringcat1.html#U3" title="RINGCAT1:sel.3">Fun</a> of <font color="Maroon" title="c1">F</font> #)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">x</font></span>)</span> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ringcat1.html#K3" title="RINGCAT1:func.3">fun</a> <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">F</font>,the <a href="ringcat1.html#U2" title="RINGCAT1:sel.2">Cod</a> of <font color="Maroon" title="c1">F</font>,the <a href="ringcat1.html#U3" title="RINGCAT1:sel.3">Fun</a> of <font color="Maroon" title="c1">F</font> #)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">y</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="grcat_1.html#D13" title="GRCAT_1:def.13">GRCAT_1:def 13</a></span>; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="grcat_1.html#D13" title="GRCAT_1:def.13">GRCAT_1:def 13</a>,<a class="ref" href="ringcat1.html#D2" target="_self" title="RINGCAT1:def.2">RINGCAT1:def 2</a>,<a class="ref" href="ringcat1.html#D6" target="_self" title="RINGCAT1:def.6">RINGCAT1:def 6</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  <a href="ringcat1.html#K3" title="RINGCAT1:func.3">fun</a> <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">F</font>,the <a href="ringcat1.html#U2" title="RINGCAT1:sel.2">Cod</a> of <font color="Maroon" title="c1">F</font>,the <a href="ringcat1.html#U3" title="RINGCAT1:sel.3">Fun</a> of <font color="Maroon" title="c1">F</font> #) is  <a href="group_6.html#V1" title="GROUP_6:attr.1">multiplicative</a>  &amp;  <a href="ringcat1.html#K3" title="RINGCAT1:func.3">fun</a> <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">F</font>,the <a href="ringcat1.html#U2" title="RINGCAT1:sel.2">Cod</a> of <font color="Maroon" title="c1">F</font>,the <a href="ringcat1.html#U3" title="RINGCAT1:sel.3">Fun</a> of <font color="Maroon" title="c1">F</font> #) is  <a href="group_1.html#V6" title="GROUP_1:attr.6">unity-preserving</a>  )</span><br/>

<span class="kw">thus </span><a NAME="E3:16"/>
for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font> being  <a href="vectsp_1.html#NM4" title="VECTSP_1:NM.4">Scalar</a> of the <a href="ringcat1.html#U1" title="RINGCAT1:sel.1">Dom</a> of <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">F</font>,the <a href="ringcat1.html#U2" title="RINGCAT1:sel.2">Cod</a> of <font color="Maroon" title="c1">F</font>,the <a href="ringcat1.html#U3" title="RINGCAT1:sel.3">Fun</a> of <font color="Maroon" title="c1">F</font> #) holds  <span class="p1">(<span class="default"><a href="ringcat1.html#K3" title="RINGCAT1:func.3">fun</a> <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">F</font>,the <a href="ringcat1.html#U2" title="RINGCAT1:sel.2">Cod</a> of <font color="Maroon" title="c1">F</font>,the <a href="ringcat1.html#U3" title="RINGCAT1:sel.3">Fun</a> of <font color="Maroon" title="c1">F</font> #)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">x</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Olive" title="b2">y</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ringcat1.html#K3" title="RINGCAT1:func.3">fun</a> <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">F</font>,the <a href="ringcat1.html#U2" title="RINGCAT1:sel.2">Cod</a> of <font color="Maroon" title="c1">F</font>,the <a href="ringcat1.html#U3" title="RINGCAT1:sel.3">Fun</a> of <font color="Maroon" title="c1">F</font> #)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">x</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ringcat1.html#K3" title="RINGCAT1:func.3">fun</a> <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">F</font>,the <a href="ringcat1.html#U2" title="RINGCAT1:sel.2">Cod</a> of <font color="Maroon" title="c1">F</font>,the <a href="ringcat1.html#U3" title="RINGCAT1:sel.3">Fun</a> of <font color="Maroon" title="c1">F</font> #)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">y</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:16"><span class="lab"><font color="Green" title="E7">A1</font></span></a>, <a class="ref" href="group_6.html#D7" title="GROUP_6:def.7">GROUP_6:def 7</a></span>; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="group_6.html#D7" title="GROUP_6:def.7">GROUP_6:def 7</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="ringcat1.html#K3" title="RINGCAT1:func.3">fun</a> <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">F</font>,the <a href="ringcat1.html#U2" title="RINGCAT1:sel.2">Cod</a> of <font color="Maroon" title="c1">F</font>,the <a href="ringcat1.html#U3" title="RINGCAT1:sel.3">Fun</a> of <font color="Maroon" title="c1">F</font> #) is  <a href="group_1.html#V6" title="GROUP_1:attr.6">unity-preserving</a> </span><br/>

<span class="kw">thus </span><a NAME="E4:16"/>
 <a href="ringcat1.html#K3" title="RINGCAT1:func.3">fun</a> <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">F</font>,the <a href="ringcat1.html#U2" title="RINGCAT1:sel.2">Cod</a> of <font color="Maroon" title="c1">F</font>,the <a href="ringcat1.html#U3" title="RINGCAT1:sel.3">Fun</a> of <font color="Maroon" title="c1">F</font> #) is  <a href="group_1.html#V6" title="GROUP_1:attr.6">unity-preserving</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:16"><span class="lab"><font color="Green" title="E7">A1</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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