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<b>let </b><font color="Maroon" title="c1">x</font>, <font color="Maroon" title="c2">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"> for <font color="Olive" title="b1">G</font> being non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>  <a href="algstr_0.html#L3" title="ALGSTR_0:struct.3">multMagma</a>  holds <br/> ( <font color="Maroon" title="c1">x</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="monoid_1.html#K9" title="MONOID_1:func.9">.:</a> <font color="Olive" title="b1">G</font>,<font color="Maroon" title="c2">X</font></span>)</span> iff <font color="Maroon" title="c1">x</font> is   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c2">X</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Olive" title="b1">G</font> )</span><br/><b>let </b><font color="Maroon" title="c3">G</font> be  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>  <a href="algstr_0.html#L3" title="ALGSTR_0:struct.3">multMagma</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c1">x</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="monoid_1.html#K9" title="MONOID_1:func.9">.:</a> <font color="Maroon" title="c3">G</font>,<font color="Maroon" title="c2">X</font></span>)</span> iff <font color="Maroon" title="c1">x</font> is   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c2">X</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">G</font> )</span><br/>





<a NAME="E1:31"/><i><font color="Green" title="E17">A1</font></i>: 
<font color="Maroon">H<sub>3</sub></font>( <a href="monoid_1.html#K9" title="MONOID_1:func.9">.:</a> <font color="Maroon" title="c3">G</font>,<font color="Maroon" title="c2">X</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="fraenkel.html#K1" title="FRAENKEL:func.1">Funcs</a> <font color="Maroon" title="c2">X</font>,<font color="Maroon">H<sub>3</sub></font>(<font color="Maroon" title="c3">G</font>)
 
<b>by </b><i><a class="ref" href="monoid_1.html#T18" target="_self" title="MONOID_1:th.18">Th18</a></i>;<br/>
<a NAME="E2:31"/>
( <font color="Maroon" title="c1">x</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="monoid_1.html#K9" title="MONOID_1:func.9">.:</a> <font color="Maroon" title="c3">G</font>,<font color="Maroon" title="c2">X</font></span>)</span> implies <font color="Maroon" title="c1">x</font> is    <a href="fraenkel.html#M2" title="FRAENKEL:mode.2">Element</a> of  <a href="fraenkel.html#K1" title="FRAENKEL:func.1">Funcs</a> <font color="Maroon" title="c2">X</font>,<font color="Maroon">H<sub>3</sub></font>(<font color="Maroon" title="c3">G</font>) )
 
<b>by </b><i><a class="ref" href="monoid_1.html#T18" target="_self" title="MONOID_1:th.18">Th18</a></i>;<br/>
<b>hence </b><a NAME="E3:31"/>
( <font color="Maroon" title="c1">x</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="monoid_1.html#K9" title="MONOID_1:func.9">.:</a> <font color="Maroon" title="c3">G</font>,<font color="Maroon" title="c2">X</font></span>)</span> implies <font color="Maroon" title="c1">x</font> is   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c2">X</font>,<font color="Maroon">H<sub>3</sub></font>(<font color="Maroon" title="c3">G</font>) )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c1">x</font> is   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c2">X</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">G</font> implies <font color="Maroon" title="c1">x</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="monoid_1.html#K9" title="MONOID_1:func.9">.:</a> <font color="Maroon" title="c3">G</font>,<font color="Maroon" title="c2">X</font></span>)</span> )</span><br/>

<b>assume </b><a NAME="E4:31"/>
<font color="Maroon" title="c1">x</font> is   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c2">X</font>,<font color="Maroon">H<sub>3</sub></font>(<font color="Maroon" title="c3">G</font>)
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c1">x</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="monoid_1.html#K9" title="MONOID_1:func.9">.:</a> <font color="Maroon" title="c3">G</font>,<font color="Maroon" title="c2">X</font></span>)</span></span><br/>

<b>then reconsider </b><font color="Maroon" title="c4">f</font> = <font color="Maroon" title="c1">x</font> as   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c2">X</font>,<font color="Maroon">H<sub>3</sub></font>(<font color="Maroon" title="c3">G</font>) ;<br/>
<a NAME="E6:31"/>
(  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c4">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">X</font> &amp;  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c4">f</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon">H<sub>3</sub></font>(<font color="Maroon" title="c3">G</font>) )
 
<b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/>
<b>hence </b><a NAME="E7:31"/>
<font color="Maroon" title="c1">x</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="monoid_1.html#K9" title="MONOID_1:func.9">.:</a> <font color="Maroon" title="c3">G</font>,<font color="Maroon" title="c2">X</font></span>)</span>
 <b>by </b><i><a class="txt" href="monoid_1.html#E1:31"><i><font color="Green" title="E17">A1</font></i></a>, <a class="ref" href="funct_2.html#D2" title="FUNCT_2:def.2">FUNCT_2:def 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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