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<b>let </b><font color="Maroon" title="c4">F</font>, <font color="Maroon" title="c5">G</font> be   <a href="msualg_3.html#NM1" title="MSUALG_3:NM.1">ManySortedFunction</a> of <font color="Maroon" title="c2">U1</font>,<span class="p1">(<span class="default"><a href="osalg_4.html#K16" title="OSALG_4:func.16">QuotOSAlg</a> <font color="Maroon" title="c2">U1</font>,<font color="Maroon" title="c3">R</font></span>)</span>; <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">s</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">S</font> holds  <font color="Maroon" title="c4">F</font> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <font color="Olive" title="b1">s</font> <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a>  <a href="osalg_4.html#K17" title="OSALG_4:func.17">OSNat_Hom</a> <font color="Maroon" title="c2">U1</font>,<font color="Maroon" title="c3">R</font>,<font color="Olive" title="b1">s</font> ) &amp; ( for <font color="Olive" title="b1">s</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">S</font> holds  <font color="Maroon" title="c5">G</font> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <font color="Olive" title="b1">s</font> <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a>  <a href="osalg_4.html#K17" title="OSALG_4:func.17">OSNat_Hom</a> <font color="Maroon" title="c2">U1</font>,<font color="Maroon" title="c3">R</font>,<font color="Olive" title="b1">s</font> ) implies <font color="Maroon" title="c4">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">G</font> )</span><br/>

<b>assume </b><b>that </b><br/><a NAME="E1:50_1_2"/><i><font color="Green" title="E25">A3</font></i>: 
for <font color="Olive" title="b1">s</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">S</font> holds  <font color="Maroon" title="c4">F</font> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <font color="Olive" title="b1">s</font> <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a>  <a href="osalg_4.html#K17" title="OSALG_4:func.17">OSNat_Hom</a> <font color="Maroon" title="c2">U1</font>,<font color="Maroon" title="c3">R</font>,<font color="Olive" title="b1">s</font>
 <b>and </b><br/><a NAME="E2:50_1_2"/><i><font color="Green" title="E26">A4</font></i>: 
for <font color="Olive" title="b1">s</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">S</font> holds  <font color="Maroon" title="c5">G</font> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <font color="Olive" title="b1">s</font> <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a>  <a href="osalg_4.html#K17" title="OSALG_4:func.17">OSNat_Hom</a> <font color="Maroon" title="c2">U1</font>,<font color="Maroon" title="c3">R</font>,<font color="Olive" title="b1">s</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c4">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">G</font></span><br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="50_1_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c6">i</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="c6">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">S</font> implies <font color="Maroon" title="c4">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">i</font> )</span><br/><b>assume </b><a NAME="E1:50_1_2_1"/>
<font color="Maroon" title="c6">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <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"> <font color="Maroon" title="c4">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">i</font></span><br/><b>then reconsider </b><font color="Maroon" title="c7">s</font> = <font color="Maroon" title="c6">i</font> as   <a href="osalg_1.html#NM1" title="OSALG_1:NM.1">SortSymbol</a> of <font color="Maroon" title="c1">S</font> ;<br/><a NAME="E3:50_1_2_1"/>
( <font color="Maroon" title="c4">F</font> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <font color="Maroon" title="c7">s</font> <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a>  <a href="osalg_4.html#K17" title="OSALG_4:func.17">OSNat_Hom</a> <font color="Maroon" title="c2">U1</font>,<font color="Maroon" title="c3">R</font>,<font color="Maroon" title="c7">s</font> &amp; <font color="Maroon" title="c5">G</font> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <font color="Maroon" title="c7">s</font> <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a>  <a href="osalg_4.html#K17" title="OSALG_4:func.17">OSNat_Hom</a> <font color="Maroon" title="c2">U1</font>,<font color="Maroon" title="c3">R</font>,<font color="Maroon" title="c7">s</font> )
 <b>by </b><i><a class="txt" href="osalg_4.html#E1:50_1_2"><i><font color="Green" title="E25">A3</font></i></a>, <a class="txt" href="osalg_4.html#E2:50_1_2"><i><font color="Green" title="E26">A4</font></i></a></i>;<br/><b>hence </b><a NAME="E4:50_1_2_1"/>
<font color="Maroon" title="c4">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">i</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div>
<b>hence </b><a NAME="E4:50_1_2"/>
<font color="Maroon" title="c4">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">G</font>
 <b>by </b><i><a class="ref" href="pboole.html#T3" title="PBOOLE:th.3">PBOOLE:3</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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