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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">U0</font> be  <a href="msafree.html#V2" title="MSAFREE:attr.2">free</a> <a href="msualg_6.html#V1" title="MSUALG_6:attr.1">feasible</a>  <a href="msualg_1.html#L3" title="MSUALG_1:struct.3">MSAlgebra</a> of <font color="Maroon" title="c1">S</font>;<br/><b>let </b><font color="Maroon" title="c3">A</font> be  <a href="msafree.html#V1" title="MSAFREE:attr.1">free</a>  <a href="msafree.html#M1" title="MSAFREE:mode.1">GeneratorSet</a> of <font color="Maroon" title="c2">U0</font>;<br/><b>let </b><font color="Maroon" title="c4">Z</font> be   <a href="msualg_2.html#NM1" title="MSUALG_2:NM.1">MSSubset</a> of <font color="Maroon" title="c2">U0</font>;<br/>







<b>assume </b><b>that </b><br/><a NAME="E1:35"/><i><font color="Green" title="E18">A1</font></i>: 
<font color="Maroon" title="c4">Z</font> <a href="pboole.html#R2" title="PBOOLE:pred.2">c=</a> <font color="Maroon" title="c3">A</font>
 <b>and </b><br/><a NAME="E2:35"/><i><font color="Green" title="E19">A2</font></i>: 
 <a href="msualg_2.html#K12" title="MSUALG_2:func.12">GenMSAlg</a> <font color="Maroon" title="c4">Z</font> is <a href="msualg_6.html#V1" title="MSUALG_6:attr.1">feasible</a>
 ;<br/>

<b>reconsider </b><font color="Maroon" title="c5">Z1</font> = <font color="Maroon" title="c4">Z</font> as   <a href="msualg_2.html#NM1" title="MSUALG_2:NM.1">MSSubset</a> of <span class="p1">(<span class="default"><a href="msualg_2.html#K12" title="MSUALG_2:func.12">GenMSAlg</a> <font color="Maroon" title="c4">Z</font></span>)</span> <b>by </b><i><a class="ref" href="msualg_2.html#D18" title="MSUALG_2:def.18">MSUALG_2:def 18</a></i>;<br/>
<a NAME="E4:35"/>
<font color="Maroon" title="c5">Z1</font> is    <a href="msafree.html#M1" title="MSAFREE:mode.1">GeneratorSet</a> of  <a href="msualg_2.html#K12" title="MSUALG_2:func.12">GenMSAlg</a> <font color="Maroon" title="c4">Z</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_1"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:35_1"/>
the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <span class="p1">(<span class="default"><a href="msualg_2.html#K12" title="MSUALG_2:func.12">GenMSAlg</a> <font color="Maroon" title="c5">Z1</font></span>)</span> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <span class="p1">(<span class="default"><a href="msualg_2.html#K12" title="MSUALG_2:func.12">GenMSAlg</a> <font color="Maroon" title="c4">Z</font></span>)</span>
 <b>by </b><i><a class="ref" href="extens_1.html#T22" title="EXTENS_1:th.22">EXTENS_1:22</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="msafree.html#D4" title="MSAFREE:def.4">MSAFREE:def 4</a><br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon" title="c6">z</font> = <font color="Maroon" title="c4">Z</font> as    <a href="msafree.html#M1" title="MSAFREE:mode.1">GeneratorSet</a> of  <a href="msualg_2.html#K12" title="MSUALG_2:func.12">GenMSAlg</a> <font color="Maroon" title="c4">Z</font> ;<br/>
<a NAME="E6:35"/>
<font color="Maroon" title="c4">Z</font> is    <a href="pboole.html#M4" title="PBOOLE:mode.4">ManySortedSubset</a> of <font color="Maroon" title="c3">A</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_2"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:35_2"/>
<font color="Maroon" title="c4">Z</font> <a href="pboole.html#R2" title="PBOOLE:pred.2">c=</a> <font color="Maroon" title="c3">A</font>
 <b>by </b><i><a class="txt" href="equation.html#E1:35"><i><font color="Green" title="E18">A1</font></i></a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="pboole.html#D23" title="PBOOLE:def.23">PBOOLE:def 23</a><br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon" title="c7">z1</font> = <font color="Maroon" title="c6">z</font> as    <a href="pboole.html#M4" title="PBOOLE:mode.4">ManySortedSubset</a> of <font color="Maroon" title="c3">A</font> ;<br/>
<b>take </b>
<font color="Maroon" title="c6">z</font>
; <i><font color="Red">:: according to </font></i><a class="ref" href="msafree.html#D6" title="MSAFREE:def.6">MSAFREE:def 6</a><br/>

<a NAME="E8:35"/>
<font color="Maroon" title="c6">z</font> is <a href="msafree.html#V1" title="MSAFREE:attr.1">free</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c8">U1</font> be  <a href="msualg_1.html#V5" title="MSUALG_1:attr.5">non-empty</a>  <a href="msualg_1.html#L3" title="MSUALG_1:struct.3">MSAlgebra</a> of <font color="Maroon" title="c1">S</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="msafree.html#D5" title="MSAFREE:def.5">MSAFREE:def 5</a><br/><b>let </b><font color="Maroon" title="c9">g</font> be    <a href="pboole.html#M3" title="PBOOLE:mode.3">ManySortedFunction</a> of <font color="Maroon" title="c6">z</font>,the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <font color="Maroon" title="c8">U1</font>;<br/>



<b>consider </b><font color="Maroon" title="c10">G</font> being    <a href="pboole.html#M3" title="PBOOLE:mode.3">ManySortedFunction</a> of <font color="Maroon" title="c3">A</font>,the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <font color="Maroon" title="c8">U1</font><b> such that </b><br/><a NAME="E2:35_3"/><i><font color="Green" title="E20">A3</font></i>: 
<font color="Maroon" title="c10">G</font> <a href="msafree.html#K1" title="MSAFREE:func.1">||</a> <font color="Maroon" title="c7">z1</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c9">g</font>
 <b>by </b><i><a class="ref" href="equation.html#T7" target="_self" title="EQUATION:th.7">Th7</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c11">h</font> being   <a href="msualg_3.html#NM1" title="MSUALG_3:NM.1">ManySortedFunction</a> of <font color="Maroon" title="c2">U0</font>,<font color="Maroon" title="c8">U1</font><b> such that </b><br/><a NAME="E4:35_3"/><i><font color="Green" title="E21">A4</font></i>: 
<font color="Maroon" title="c11">h</font> <a href="msualg_3.html#R1" title="MSUALG_3:pred.1">is_homomorphism</a> <font color="Maroon" title="c2">U0</font>,<font color="Maroon" title="c8">U1</font>
 <b>and </b><br/><a NAME="E5:35_3"/><i><font color="Green" title="E22">A5</font></i>: 
<font color="Maroon" title="c11">h</font> <a href="msafree.html#K1" title="MSAFREE:func.1">||</a> <font color="Maroon" title="c3">A</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c10">G</font>
 <b>by </b><i><a class="ref" href="msafree.html#D5" title="MSAFREE:def.5">MSAFREE:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c12">D</font> = the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <span class="p1">(<span class="default"><a href="msualg_2.html#K12" title="MSUALG_2:func.12">GenMSAlg</a> <font color="Maroon" title="c4">Z</font></span>)</span> as   <a href="msualg_2.html#NM1" title="MSUALG_2:NM.1">MSSubset</a> of <font color="Maroon" title="c2">U0</font> <b>by </b><i><a class="ref" href="msualg_2.html#D10" title="MSUALG_2:def.10">MSUALG_2:def 10</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c13">H</font> = <font color="Maroon" title="c11">h</font> <a href="msafree.html#K1" title="MSAFREE:func.1">||</a> <font color="Maroon" title="c12">D</font> as   <a href="msualg_3.html#NM1" title="MSUALG_3:NM.1">ManySortedFunction</a> of <span class="p1">(<span class="default"><a href="msualg_2.html#K12" title="MSUALG_2:func.12">GenMSAlg</a> <font color="Maroon" title="c4">Z</font></span>)</span>,<font color="Maroon" title="c8">U1</font> ;<br/>
<b>take </b>
<font color="Maroon" title="c13">H</font>
;<br/>

<b>thus </b><a NAME="E8:35_3"/>
<font color="Maroon" title="c13">H</font> <a href="msualg_3.html#R1" title="MSUALG_3:pred.1">is_homomorphism</a>  <a href="msualg_2.html#K12" title="MSUALG_2:func.12">GenMSAlg</a> <font color="Maroon" title="c4">Z</font>,<font color="Maroon" title="c8">U1</font>
 <b>by </b><i><a class="txt" href="equation.html#E2:35"><i><font color="Green" title="E19">A2</font></i></a>, <a class="txt" href="equation.html#E4:35_3"><i><font color="Green" title="E21">A4</font></i></a>, <a class="ref" href="equation.html#T24" target="_self" title="EQUATION:th.24">Th24</a></i>;<br/>

<b>thus </b><font color="Maroon" title="c9">g</font> = 
<font color="Maroon" title="c11">h</font> <a href="msafree.html#K1" title="MSAFREE:func.1">||</a> <font color="Maroon" title="c4">Z</font>
<b>by </b><i><a class="txt" href="equation.html#E2:35_3"><i><font color="Green" title="E20">A3</font></i></a>, <a class="txt" href="equation.html#E5:35_3"><i><font color="Green" title="E22">A5</font></i></a>, <a class="ref" href="equation.html#T6" target="_self" title="EQUATION:th.6">Th6</a></i>
<br/>.= 
<font color="Maroon" title="c13">H</font> <a href="msafree.html#K1" title="MSAFREE:func.1">||</a> <font color="Maroon" title="c6">z</font>
<b>by </b><i><a class="ref" href="equation.html#T6" target="_self" title="EQUATION:th.6">Th6</a></i>
;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E9:35"/>
<font color="Maroon" title="c6">z</font> is <a href="msafree.html#V1" title="MSAFREE:attr.1">free</a>
 ;<br/>


</div>
