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<b>let </b><font color="Maroon" title="c1">S</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="c1">S</font> is <a href="card_3.html#V3" title="CARD_3:attr.3">product-like</a> implies ( <font color="Maroon" title="c1">S</font> is <a href="fraenkel.html#V1" title="FRAENKEL:attr.1">functional</a> &amp; <font color="Maroon" title="c1">S</font> is <a href="card_3.html#V2" title="CARD_3:attr.2">with_common_domain</a> ) )</span><br/>

<b>given </b><font color="Maroon" title="c2">f</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><b> such that </b><a NAME="E2:102_1_1"/><i><font color="Green" title="E52">A1</font></i>: 
<font color="Maroon" title="c1">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <font color="Maroon" title="c2">f</font>
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="card_3.html#D14" target="_self" title="CARD_3:def.14">CARD_3:def 14</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">S</font> is <a href="fraenkel.html#V1" title="FRAENKEL:attr.1">functional</a> &amp; <font color="Maroon" title="c1">S</font> is <a href="card_3.html#V2" title="CARD_3:attr.2">with_common_domain</a> )</span><br/>

<b>thus </b><a NAME="E3:102_1_1"/>
<font color="Maroon" title="c1">S</font> is <a href="fraenkel.html#V1" title="FRAENKEL:attr.1">functional</a>
 <b>by </b><i><a class="txt" href="#E2:102_1_1"><i><font color="Green" title="E52">A1</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c1">S</font> is <a href="card_3.html#V2" title="CARD_3:attr.2">with_common_domain</a></span><br/>

<b>let </b><font color="Maroon" title="c3">h</font>, <font color="Maroon" title="c4">g</font> be   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a>; <i><font color="Red">:: according to </font></i><a class="ref" href="card_3.html#D10" target="_self" title="CARD_3:def.10">CARD_3:def 10</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c3">h</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">S</font> &amp; <font color="Maroon" title="c4">g</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">S</font> implies  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c3">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c4">g</font> )</span><br/>

<b>assume </b><b>that </b><br/><a NAME="E4:102_1_1"/><i><font color="Green" title="E53">A2</font></i>: 
<font color="Maroon" title="c3">h</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">S</font>
 <b>and </b><br/><a NAME="E5:102_1_1"/><i><font color="Green" title="E54">A3</font></i>: 
<font color="Maroon" title="c4">g</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">S</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c3">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c4">g</font></span><br/>

<b>thus </b> <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c3">h</font> = 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c2">f</font>
<b>by </b><i><a class="txt" href="#E2:102_1_1"><i><font color="Green" title="E52">A1</font></i></a>, <a class="txt" href="#E4:102_1_1"><i><font color="Green" title="E53">A2</font></i></a>, <a class="ref" href="card_3.html#T18" target="_self" title="CARD_3:th.18">Th18</a></i>
<br/>.= 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c4">g</font>
<b>by </b><i><a class="txt" href="#E2:102_1_1"><i><font color="Green" title="E52">A1</font></i></a>, <a class="txt" href="#E5:102_1_1"><i><font color="Green" title="E54">A3</font></i></a>, <a class="ref" href="card_3.html#T18" target="_self" title="CARD_3:th.18">Th18</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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