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<b>let </b><font color="Maroon" title="c1">I</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>let </b><font color="Maroon" title="c2">S</font> be  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> non <a href="msualg_1.html#V2" title="MSUALG_1:attr.2">void</a>  <a href="msualg_1.html#L1" title="MSUALG_1:struct.1">ManySortedSign</a> ;<br/><b>let </b><font color="Maroon" title="c3">E</font> be   <a href="equation.html#NM1" title="EQUATION:NM.1">EqualSet</a> of <font color="Maroon" title="c2">S</font>;<br/><b>let </b><font color="Maroon" title="c4">F</font> be    <a href="pralg_2.html#M2" title="PRALG_2:mode.2">MSAlgebra-Family</a> of <font color="Maroon" title="c1">I</font>,<font color="Maroon" title="c2">S</font>;<br/>







<b>assume </b><a NAME="E1:55"/><i><font color="Green" title="E27">A1</font></i>: 
for <font color="Olive" title="b1">i</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">I</font> holds <br/> ex <font color="Olive" title="b2">A</font> being   <a href="msualg_1.html#L3" title="MSUALG_1:struct.3">MSAlgebra</a> of <font color="Maroon" title="c2">S</font> st <br/>( <font color="Olive" title="b2">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font> &amp; <font color="Olive" title="b2">A</font> <a href="equation.html#R2" title="EQUATION:pred.2">|=</a> <font color="Maroon" title="c3">E</font> )
 ;<br/>

<b>let </b><font color="Maroon" title="c5">s</font> be   <a href="msualg_1.html#NM1" title="MSUALG_1:NM.1">SortSymbol</a> of <font color="Maroon" title="c2">S</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="equation.html#D6" target="_self" title="EQUATION:def.6">EQUATION:def 6</a><br/><b>let </b><font color="Maroon" title="c6">e</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1">(<span class="default"><a href="equation.html#K4" title="EQUATION:func.4">Equations</a> <font color="Maroon" title="c2">S</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">s</font>;<br/>



<b>assume </b><a NAME="E2:55"/><i><font color="Green" title="E28">A2</font></i>: 
<font color="Maroon" title="c6">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">E</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">s</font>
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="55_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c7">i</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:55_1"/>
<font color="Maroon" title="c7">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">I</font>
 ;<br/><b>then consider </b><font color="Maroon" title="c8">A</font> being    <a href="msualg_1.html#L3" title="MSUALG_1:struct.3">MSAlgebra</a> of <font color="Maroon" title="c2">S</font><b> such that </b><br/><a NAME="E3:55_1"/><i><font color="Green" title="E29">A3</font></i>: 
( <font color="Maroon" title="c8">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">i</font> &amp; <font color="Maroon" title="c8">A</font> <a href="equation.html#R2" title="EQUATION:pred.2">|=</a> <font color="Maroon" title="c3">E</font> )
 <b>by </b><i><a class="txt" href="equation.html#E1:55"><i><font color="Green" title="E27">A1</font></i></a></i>;<br/><b>take </b><font color="Maroon" title="c9">A</font> = <font color="Maroon" title="c8">A</font>;<br/><b>thus </b><a NAME="E4:55_1"/>
( <font color="Maroon" title="c9">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">i</font> &amp; <font color="Maroon" title="c9">A</font> <a href="equation.html#R1" title="EQUATION:pred.1">|=</a> <font color="Maroon" title="c6">e</font> )
 <b>by </b><i><a class="txt" href="equation.html#E2:55"><i><font color="Green" title="E28">A2</font></i></a>, <a class="txt" href="equation.html#E3:55_1"><i><font color="Green" title="E29">A3</font></i></a>, <a class="ref" href="equation.html#D6" target="_self" title="EQUATION:def.6">Def6</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E4:55"/>
 <a href="pralg_2.html#K15" title="PRALG_2:func.15">product</a> <font color="Maroon" title="c4">F</font> <a href="equation.html#R1" title="EQUATION:pred.1">|=</a> <font color="Maroon" title="c6">e</font>
 <b>by </b><i><a class="ref" href="equation.html#T39" target="_self" title="EQUATION:th.39">Th39</a></i>;<br/>


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