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<b>let </b><font color="Maroon" title="c1">D</font> be  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>let </b><font color="Maroon" title="c2">G</font>, <font color="Maroon" title="c3">H</font> be   <a href="funcsdom.html#NM1" title="FUNCSDOM:NM.1">Ring</a>;<br/>





<b>thus </b><a NAME="E1:47"/>
( <font color="Maroon" title="c1">D</font> is    <a href="ringcat1.html#M4" title="RINGCAT1:mode.4">RingMorphism_DOMAIN</a> of <font color="Maroon" title="c2">G</font>,<font color="Maroon" title="c3">H</font> implies for <font color="Olive" title="b1">x</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">D</font> holds  <font color="Olive" title="b1">x</font> is    <a href="ringcat1.html#M1" title="RINGCAT1:mode.1">Morphism</a> of <font color="Maroon" title="c2">G</font>,<font color="Maroon" title="c3">H</font> )
 <b>by </b><i><a class="ref" href="ringcat1.html#D14" target="_self" title="RINGCAT1:def.14">Def14</a></i>;<br/>

<b>thus </b><a NAME="E2:47"/>
( ( for <font color="Olive" title="b1">x</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">D</font> holds  <font color="Olive" title="b1">x</font> is    <a href="ringcat1.html#M1" title="RINGCAT1:mode.1">Morphism</a> of <font color="Maroon" title="c2">G</font>,<font color="Maroon" title="c3">H</font> ) implies <font color="Maroon" title="c1">D</font> is    <a href="ringcat1.html#M4" title="RINGCAT1:mode.4">RingMorphism_DOMAIN</a> of <font color="Maroon" title="c2">G</font>,<font color="Maroon" title="c3">H</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="47_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:47_1"/><i><font color="Green" title="E30">A1</font></i>: 
for <font color="Olive" title="b1">x</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">D</font> holds  <font color="Olive" title="b1">x</font> is    <a href="ringcat1.html#M1" title="RINGCAT1:mode.1">Morphism</a> of <font color="Maroon" title="c2">G</font>,<font color="Maroon" title="c3">H</font>
 ;<br/>

<a NAME="E2:47_1"/><b>then </b>
for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">D</font> holds <br/><font color="Olive" title="b1">x</font> is  <a href="ringcat1.html#V2" title="RINGCAT1:attr.2">strict</a> <a href="ringcat1.html#NM1" title="RINGCAT1:NM.1">RingMorphism</a>
 
;<br/>
<b>then reconsider </b><font color="Maroon" title="c4">D'</font> = <font color="Maroon" title="c1">D</font> as   <a href="ringcat1.html#NM5" title="RINGCAT1:NM.5">RingMorphism_DOMAIN</a> <b>by </b><i><a class="ref" href="ringcat1.html#D13" target="_self" title="RINGCAT1:def.13">Def13</a></i>;<br/>
<a NAME="E4:47_1"/>
for <font color="Olive" title="b1">x</font> being   <a href="ringcat1.html#M3" title="RINGCAT1:mode.3">Element</a> of <font color="Maroon" title="c4">D'</font> holds  <font color="Olive" title="b1">x</font> is    <a href="ringcat1.html#M1" title="RINGCAT1:mode.1">Morphism</a> of <font color="Maroon" title="c2">G</font>,<font color="Maroon" title="c3">H</font>
 
<b>by </b><i><a class="txt" href="ringcat1.html#E1:47_1"><i><font color="Green" title="E30">A1</font></i></a></i>;<br/>
<b>hence </b><a NAME="E5:47_1"/>
<font color="Maroon" title="c1">D</font> is    <a href="ringcat1.html#M4" title="RINGCAT1:mode.4">RingMorphism_DOMAIN</a> of <font color="Maroon" title="c2">G</font>,<font color="Maroon" title="c3">H</font>
 <b>by </b><i><a class="ref" href="ringcat1.html#D14" target="_self" title="RINGCAT1:def.14">Def14</a></i>;<br/>


</div><b>end;</b></div>


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