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<b>let </b><font color="Maroon" title="c1">S</font> be  non <a href="struct_0.html#V3" title="STRUCT_0:attr.3">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="c2">s</font> be   <a href="msualg_1.html#NM1" title="MSUALG_1:NM.1">SortSymbol</a> of <font color="Maroon" title="c1">S</font>;<br/><b>let </b><font color="Maroon" title="c3">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="c1">S</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c2">s</font>;<br/><b>let </b><font color="Maroon" title="c4">I</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="c5">A</font> be    <a href="pralg_2.html#M2" title="PRALG_2:mode.2">MSAlgebra-Family</a> of <font color="Maroon" title="c4">I</font>,<font color="Maroon" title="c1">S</font>;<br/>









<b>assume </b><a NAME="E1:53"/><i><font color="Green" title="E26">A1</font></i>: 
for <font color="Olive" title="b1">i</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c4">I</font> holds  <font color="Maroon" title="c5">A</font> <a href="pralg_2.html#K6" title="PRALG_2:func.6">.</a> <font color="Olive" title="b1">i</font> <a href="equation.html#R1" title="EQUATION:pred.1">|=</a> <font color="Maroon" title="c3">e</font>
 ;<br/>

<b>let </b><font color="Maroon" title="c6">h</font> be   <a href="msualg_3.html#NM1" title="MSUALG_3:NM.1">ManySortedFunction</a> of <span class="p1">(<span class="default"><a href="equation.html#K3" title="EQUATION:func.3">TermAlg</a> <font color="Maroon" title="c1">S</font></span>)</span>,<span class="p1">(<span class="default"><a href="pralg_2.html#K15" title="PRALG_2:func.15">product</a> <font color="Maroon" title="c5">A</font></span>)</span>; <i><font color="Red">:: according to </font></i><a class="ref" href="equation.html#D5" target="_self" title="EQUATION:def.5">EQUATION:def 5</a><br/>

<b>assume </b><a NAME="E2:53"/><i><font color="Green" title="E27">A2</font></i>: 
<font color="Maroon" title="c6">h</font> <a href="msualg_3.html#R1" title="MSUALG_3:pred.1">is_homomorphism</a>  <a href="equation.html#K3" title="EQUATION:func.3">TermAlg</a> <font color="Maroon" title="c1">S</font>, <a href="pralg_2.html#K15" title="PRALG_2:func.15">product</a> <font color="Maroon" title="c5">A</font>
 ;<br/>

<a NAME="E3:53"/><i><font color="Green" title="E28">A3</font></i>: 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">h</font> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <font color="Maroon" title="c2">s</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <span class="p1">(<span class="default"><a href="equation.html#K3" title="EQUATION:func.3">TermAlg</a> <font color="Maroon" title="c1">S</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c2">s</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>reconsider </b><font color="Maroon" title="c7">e1</font> = <font color="Maroon" title="c3">e</font> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a>  as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <span class="p1">(<span class="default"><a href="equation.html#K3" title="EQUATION:func.3">TermAlg</a> <font color="Maroon" title="c1">S</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c2">s</font> <b>by </b><i><a class="ref" href="equation.html#T31" target="_self" title="EQUATION:th.31">Th31</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c8">e2</font> = <font color="Maroon" title="c3">e</font> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a>  as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <span class="p1">(<span class="default"><a href="equation.html#K3" title="EQUATION:func.3">TermAlg</a> <font color="Maroon" title="c1">S</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c2">s</font> <b>by </b><i><a class="ref" href="equation.html#T32" target="_self" title="EQUATION:th.32">Th32</a></i>;<br/>
<a NAME="E6:53"/><i><font color="Green" title="E29">A4</font></i>: 
the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <span class="p1">(<span class="default"><a href="pralg_2.html#K15" title="PRALG_2:func.15">product</a> <font color="Maroon" title="c5">A</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c2">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> <span class="p1">(<span class="default"><a href="pralg_2.html#K10" title="PRALG_2:func.10">Carrier</a> <font color="Maroon" title="c5">A</font>,<font color="Maroon" title="c2">s</font></span>)</span>
 
<b>by </b><i><a class="ref" href="pralg_2.html#D17" title="PRALG_2:def.17">PRALG_2:def 17</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="53_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c9">i</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c4">I</font>;<br/><b>set </b><font color="Maroon" title="c10">Z</font> = <span class="p1">(<span class="default"><a href="pralg_3.html#K4" title="PRALG_3:func.4">proj</a> <font color="Maroon" title="c5">A</font>,<font color="Maroon" title="c9">i</font></span>)</span> <a href="msualg_3.html#K3" title="MSUALG_3:func.3">**</a> <font color="Maroon" title="c6">h</font>;<br/><a NAME="E1:53_1"/><i><font color="Green" title="E30">A5</font></i>: 
<font color="Maroon" title="c5">A</font> <a href="pralg_2.html#K6" title="PRALG_2:func.6">.</a> <font color="Maroon" title="c9">i</font> <a href="equation.html#R1" title="EQUATION:pred.1">|=</a> <font color="Maroon" title="c3">e</font>
 <b>by </b><i><a class="txt" href="equation.html#E1:53"><i><font color="Green" title="E26">A1</font></i></a></i>;<br/><a NAME="E2:53_1"/>
 <a href="pralg_3.html#K4" title="PRALG_3:func.4">proj</a> <font color="Maroon" title="c5">A</font>,<font color="Maroon" title="c9">i</font> <a href="msualg_3.html#R1" title="MSUALG_3:pred.1">is_homomorphism</a>  <a href="pralg_2.html#K15" title="PRALG_2:func.15">product</a> <font color="Maroon" title="c5">A</font>,<font color="Maroon" title="c5">A</font> <a href="pralg_2.html#K6" title="PRALG_2:func.6">.</a> <font color="Maroon" title="c9">i</font>
 <b>by </b><i><a class="ref" href="pralg_3.html#T25" title="PRALG_3:th.25">PRALG_3:25</a></i>;<br/><a NAME="E3:53_1"/><b>then </b><i><font color="Green" title="E31">A6</font></i>: 
<span class="p1">(<span class="default"><a href="pralg_3.html#K4" title="PRALG_3:func.4">proj</a> <font color="Maroon" title="c5">A</font>,<font color="Maroon" title="c9">i</font></span>)</span> <a href="msualg_3.html#K3" title="MSUALG_3:func.3">**</a> <font color="Maroon" title="c6">h</font> <a href="msualg_3.html#R1" title="MSUALG_3:pred.1">is_homomorphism</a>  <a href="equation.html#K3" title="EQUATION:func.3">TermAlg</a> <font color="Maroon" title="c1">S</font>,<font color="Maroon" title="c5">A</font> <a href="pralg_2.html#K6" title="PRALG_2:func.6">.</a> <font color="Maroon" title="c9">i</font>
 <b>by </b><i><a class="txt" href="equation.html#E2:53"><i><font color="Green" title="E27">A2</font></i></a>, <a class="ref" href="msualg_3.html#T10" title="MSUALG_3:th.10">MSUALG_3:10</a></i>;<br/><b>thus </b><span class="p1">(<span class="default"><a href="pralg_3.html#K3" title="PRALG_3:func.3">proj</a> <span class="p2">(<span class="default"><a href="pralg_2.html#K10" title="PRALG_2:func.10">Carrier</a> <font color="Maroon" title="c5">A</font>,<font color="Maroon" title="c2">s</font></span>)</span>,<font color="Maroon" title="c9">i</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">h</font> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <font color="Maroon" title="c2">s</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">e1</font></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pralg_3.html#K4" title="PRALG_3:func.4">proj</a> <font color="Maroon" title="c5">A</font>,<font color="Maroon" title="c9">i</font></span>)</span> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <font color="Maroon" title="c2">s</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">h</font> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <font color="Maroon" title="c2">s</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">e1</font></span>)</span>
<b>by </b><i><a class="ref" href="pralg_3.html#D3" title="PRALG_3:def.3">PRALG_3:def 3</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pralg_3.html#K4" title="PRALG_3:func.4">proj</a> <font color="Maroon" title="c5">A</font>,<font color="Maroon" title="c9">i</font></span>)</span> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <font color="Maroon" title="c2">s</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">h</font> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <font color="Maroon" title="c2">s</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">e1</font>
<b>by </b><i><a class="txt" href="equation.html#E3:53"><i><font color="Green" title="E28">A3</font></i></a>, <a class="ref" href="funct_1.html#T23" title="FUNCT_1:th.23">FUNCT_1:23</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pralg_3.html#K4" title="PRALG_3:func.4">proj</a> <font color="Maroon" title="c5">A</font>,<font color="Maroon" title="c9">i</font></span>)</span> <a href="msualg_3.html#K3" title="MSUALG_3:func.3">**</a> <font color="Maroon" title="c6">h</font></span>)</span> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <font color="Maroon" title="c2">s</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">e1</font>
<b>by </b><i><a class="ref" href="msualg_3.html#T2" title="MSUALG_3:th.2">MSUALG_3:2</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pralg_3.html#K4" title="PRALG_3:func.4">proj</a> <font color="Maroon" title="c5">A</font>,<font color="Maroon" title="c9">i</font></span>)</span> <a href="msualg_3.html#K3" title="MSUALG_3:func.3">**</a> <font color="Maroon" title="c6">h</font></span>)</span> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <font color="Maroon" title="c2">s</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">e2</font>
<b>by </b><i><a class="txt" href="equation.html#E1:53_1"><i><font color="Green" title="E30">A5</font></i></a>, <a class="txt" href="equation.html#E3:53_1"><i><font color="Green" title="E31">A6</font></i></a>, <a class="ref" href="equation.html#D5" target="_self" title="EQUATION:def.5">Def5</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pralg_3.html#K4" title="PRALG_3:func.4">proj</a> <font color="Maroon" title="c5">A</font>,<font color="Maroon" title="c9">i</font></span>)</span> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <font color="Maroon" title="c2">s</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">h</font> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <font color="Maroon" title="c2">s</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">e2</font>
<b>by </b><i><a class="ref" href="msualg_3.html#T2" title="MSUALG_3:th.2">MSUALG_3:2</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pralg_3.html#K4" title="PRALG_3:func.4">proj</a> <font color="Maroon" title="c5">A</font>,<font color="Maroon" title="c9">i</font></span>)</span> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <font color="Maroon" title="c2">s</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">h</font> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <font color="Maroon" title="c2">s</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">e2</font></span>)</span>
<b>by </b><i><a class="txt" href="equation.html#E3:53"><i><font color="Green" title="E28">A3</font></i></a>, <a class="ref" href="funct_1.html#T23" title="FUNCT_1:th.23">FUNCT_1:23</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="pralg_3.html#K3" title="PRALG_3:func.3">proj</a> <span class="p2">(<span class="default"><a href="pralg_2.html#K10" title="PRALG_2:func.10">Carrier</a> <font color="Maroon" title="c5">A</font>,<font color="Maroon" title="c2">s</font></span>)</span>,<font color="Maroon" title="c9">i</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">h</font> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <font color="Maroon" title="c2">s</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">e2</font></span>)</span>
<b>by </b><i><a class="ref" href="pralg_3.html#D3" title="PRALG_3:def.3">PRALG_3:def 3</a></i>
;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E8:53"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c6">h</font> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <font color="Maroon" title="c2">s</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">e</font> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">h</font> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <font color="Maroon" title="c2">s</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">e</font> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> </span>)</span>
 <b>by </b><i><a class="txt" href="equation.html#E6:53"><i><font color="Green" title="E29">A4</font></i></a>, <a class="ref" href="msualg_9.html#T15" title="MSUALG_9:th.15">MSUALG_9:15</a></i>;<br/>


</div>
