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<b>let </b><font color="Maroon" title="c2">D1</font>, <font color="Maroon" title="c3">D2</font> be   <a href="ringcat1.html#NM5" title="RINGCAT1:NM.5">RingMorphism_DOMAIN</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( ( for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  holds <br/> ( <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">D1</font> iff  ex <font color="Olive" title="b2">G</font>, <font color="Olive" title="b3">H</font> being   <a href="ringcat1.html#M2" title="RINGCAT1:mode.2">Element</a> of <font color="Maroon" title="c1">V</font> st <br/>( <font color="Olive" title="b2">G</font> <a href="ringcat1.html#R1" title="RINGCAT1:pred.1">&lt;=</a> <font color="Olive" title="b3">H</font> &amp; <font color="Olive" title="b1">x</font> is    <a href="ringcat1.html#M1" title="RINGCAT1:mode.1">Morphism</a> of <font color="Olive" title="b2">G</font>,<font color="Olive" title="b3">H</font> ) ) ) &amp; ( for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  holds <br/> ( <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">D2</font> iff  ex <font color="Olive" title="b2">G</font>, <font color="Olive" title="b3">H</font> being   <a href="ringcat1.html#M2" title="RINGCAT1:mode.2">Element</a> of <font color="Maroon" title="c1">V</font> st <br/>( <font color="Olive" title="b2">G</font> <a href="ringcat1.html#R1" title="RINGCAT1:pred.1">&lt;=</a> <font color="Olive" title="b3">H</font> &amp; <font color="Olive" title="b1">x</font> is    <a href="ringcat1.html#M1" title="RINGCAT1:mode.1">Morphism</a> of <font color="Olive" title="b2">G</font>,<font color="Olive" title="b3">H</font> ) ) ) implies <font color="Maroon" title="c2">D1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">D2</font> )</span><br/>

<b>assume </b><b>that </b><br/><a NAME="E1:61_1_2"/><i><font color="Green" title="E37">A11</font></i>: 
for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  holds <br/> ( <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">D1</font> iff  ex <font color="Olive" title="b2">G</font>, <font color="Olive" title="b3">H</font> being   <a href="ringcat1.html#M2" title="RINGCAT1:mode.2">Element</a> of <font color="Maroon" title="c1">V</font> st <br/>( <font color="Olive" title="b2">G</font> <a href="ringcat1.html#R1" title="RINGCAT1:pred.1">&lt;=</a> <font color="Olive" title="b3">H</font> &amp; <font color="Olive" title="b1">x</font> is    <a href="ringcat1.html#M1" title="RINGCAT1:mode.1">Morphism</a> of <font color="Olive" title="b2">G</font>,<font color="Olive" title="b3">H</font> ) )
 <b>and </b><br/><a NAME="E2:61_1_2"/><i><font color="Green" title="E38">A12</font></i>: 
for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  holds <br/> ( <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">D2</font> iff  ex <font color="Olive" title="b2">G</font>, <font color="Olive" title="b3">H</font> being   <a href="ringcat1.html#M2" title="RINGCAT1:mode.2">Element</a> of <font color="Maroon" title="c1">V</font> st <br/>( <font color="Olive" title="b2">G</font> <a href="ringcat1.html#R1" title="RINGCAT1:pred.1">&lt;=</a> <font color="Olive" title="b3">H</font> &amp; <font color="Olive" title="b1">x</font> is    <a href="ringcat1.html#M1" title="RINGCAT1:mode.1">Morphism</a> of <font color="Olive" title="b2">G</font>,<font color="Olive" title="b3">H</font> ) )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c2">D1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">D2</font></span><br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="61_1_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c4">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">D1</font> iff <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">D2</font> )</span><br/><a NAME="E1:61_1_2_1"/>
( <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">D1</font> iff  ex <font color="Olive" title="b1">G</font>, <font color="Olive" title="b2">H</font> being   <a href="ringcat1.html#M2" title="RINGCAT1:mode.2">Element</a> of <font color="Maroon" title="c1">V</font> st <br/>( <font color="Olive" title="b1">G</font> <a href="ringcat1.html#R1" title="RINGCAT1:pred.1">&lt;=</a> <font color="Olive" title="b2">H</font> &amp; <font color="Maroon" title="c4">x</font> is    <a href="ringcat1.html#M1" title="RINGCAT1:mode.1">Morphism</a> of <font color="Olive" title="b1">G</font>,<font color="Olive" title="b2">H</font> ) )
 <b>by </b><i><a class="txt" href="ringcat1.html#E1:61_1_2"><i><font color="Green" title="E37">A11</font></i></a></i>;<br/><b>hence </b><a NAME="E2:61_1_2_1"/>
( <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">D1</font> iff <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">D2</font> )
 <b>by </b><i><a class="txt" href="ringcat1.html#E2:61_1_2"><i><font color="Green" title="E38">A12</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div>
<b>hence </b><a NAME="E4:61_1_2"/>
<font color="Maroon" title="c2">D1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">D2</font>
 <b>by </b><i><a class="ref" href="tarski.html#T2" title="TARSKI:th.2">TARSKI:2</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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