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<span class="kw">defpred </span><font color="Maroon">S<sub>1</sub></font>[  <a href="ringcat1.html#NM1" title="RINGCAT1:NM.1">RingMorphism</a>,  <a href="ringcat1.html#NM1" title="RINGCAT1:NM.1">RingMorphism</a>] <span class="kw">means </span> <a href="ringcat1.html#K1" title="RINGCAT1:func.1">dom</a> $1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ringcat1.html#K2" title="RINGCAT1:func.2">cod</a> $2;<br/>
<span class="kw">let </span><font color="Maroon" title="c1">g</font>, <font color="Maroon" title="c2">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#K1" title="RINGCAT1:func.1">dom</a> <font color="Maroon" title="c1">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ringcat1.html#K2" title="RINGCAT1:func.2">cod</a> <font color="Maroon" title="c2">f</font> implies  ex <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> st <br/>( <font color="Olive" title="b1">G1</font> <a href="ringcat1.html#R1" title="RINGCAT1:pred.1">&lt;=</a> <font color="Olive" title="b2">G2</font> &amp; <font color="Olive" title="b2">G2</font> <a href="ringcat1.html#R1" title="RINGCAT1:pred.1">&lt;=</a> <font color="Olive" title="b3">G3</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="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="Olive" title="b2">G2</font>,<font color="Olive" title="b3">G3</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> #) is    <a href="ringcat1.html#M1" title="RINGCAT1:mode.1">Morphism</a> of <font color="Olive" title="b1">G1</font>,<font color="Olive" title="b2">G2</font> ) )</span><br/>

<span class="kw">assume </span><a NAME="E1:25"/><span class="lab"><font color="Green" title="E13">A1</font></span>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c1">g</font>,<font color="Maroon" title="c2">f</font>]
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <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> st <br/>( <font color="Olive" title="b1">G1</font> <a href="ringcat1.html#R1" title="RINGCAT1:pred.1">&lt;=</a> <font color="Olive" title="b2">G2</font> &amp; <font color="Olive" title="b2">G2</font> <a href="ringcat1.html#R1" title="RINGCAT1:pred.1">&lt;=</a> <font color="Olive" title="b3">G3</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="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="Olive" title="b2">G2</font>,<font color="Olive" title="b3">G3</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> #) is    <a href="ringcat1.html#M1" title="RINGCAT1:mode.1">Morphism</a> of <font color="Olive" title="b1">G1</font>,<font color="Olive" title="b2">G2</font> )</span><br/>

<a NAME="E2:25"/>
(  ex <font color="Olive" title="b1">G2</font>, <font color="Olive" title="b2">G3</font> being  <a href="funcsdom.html#NM1" title="FUNCSDOM:NM.1">Ring</a> st <br/>( <font color="Olive" title="b1">G2</font> <a href="ringcat1.html#R1" title="RINGCAT1:pred.1">&lt;=</a> <font color="Olive" title="b2">G3</font> &amp;  <a href="ringcat1.html#K1" title="RINGCAT1:func.1">dom</a> <font color="Maroon" title="c1">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">G2</font> &amp;  <a href="ringcat1.html#K2" title="RINGCAT1:func.2">cod</a> <font color="Maroon" title="c1">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">G3</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="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="Olive" title="b1">G2</font>,<font color="Olive" title="b2">G3</font> ) &amp;  ex <font color="Olive" title="b1">G1</font>, <font color="Olive" title="b2">G29</font> being  <a href="funcsdom.html#NM1" title="FUNCSDOM:NM.1">Ring</a> st <br/>( <font color="Olive" title="b1">G1</font> <a href="ringcat1.html#R1" title="RINGCAT1:pred.1">&lt;=</a> <font color="Olive" title="b2">G29</font> &amp;  <a href="ringcat1.html#K1" title="RINGCAT1:func.1">dom</a> <font color="Maroon" title="c2">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">G1</font> &amp;  <a href="ringcat1.html#K2" title="RINGCAT1:func.2">cod</a> <font color="Maroon" title="c2">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">G29</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> #) is    <a href="ringcat1.html#M1" title="RINGCAT1:mode.1">Morphism</a> of <font color="Olive" title="b1">G1</font>,<font color="Olive" title="b2">G29</font> ) )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="ringcat1.html#E18"><span class="lab"><font color="Green" title="E11">Lm6</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E3:25"/>
 ex <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> st <br/>( <font color="Olive" title="b1">G1</font> <a href="ringcat1.html#R1" title="RINGCAT1:pred.1">&lt;=</a> <font color="Olive" title="b2">G2</font> &amp; <font color="Olive" title="b2">G2</font> <a href="ringcat1.html#R1" title="RINGCAT1:pred.1">&lt;=</a> <font color="Olive" title="b3">G3</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="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="Olive" title="b2">G2</font>,<font color="Olive" title="b3">G3</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> #) is    <a href="ringcat1.html#M1" title="RINGCAT1:mode.1">Morphism</a> of <font color="Olive" title="b1">G1</font>,<font color="Olive" title="b2">G2</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:25"><span class="lab"><font color="Green" title="E13">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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