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<b>let </b><font color="Maroon" title="c1">S</font> be  <a href="osalg_4.html#V2" title="OSALG_4:attr.2">locally_directed</a> <a href="osalg_1.html#NM3" title="OSALG_1:NM.3">OrderSortedSign</a>; <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">U1</font>, <font color="Olive" title="b2">U2</font> being <a href="msualg_1.html#V5" title="MSUALG_1:attr.5">non-empty</a> <a href="osalg_1.html#NM5" title="OSALG_1:NM.5">OSAlgebra</a> of <font color="Maroon" title="c1">S</font><br/> for <font color="Olive" title="b3">F</font> being  <a href="msualg_3.html#NM1" title="MSUALG_3:NM.1">ManySortedFunction</a> of <font color="Olive" title="b1">U1</font>,<font color="Olive" title="b2">U2</font>  st <font color="Olive" title="b3">F</font> <a href="msualg_3.html#R1" title="MSUALG_3:pred.1">is_homomorphism</a> <font color="Olive" title="b1">U1</font>,<font color="Olive" title="b2">U2</font> &amp; <font color="Olive" title="b3">F</font> is <a href="osalg_3.html#V1" title="OSALG_3:attr.1">order-sorted</a> holds <br/> <a href="msualg_4.html#K18" title="MSUALG_4:func.18">MSCng</a> <font color="Olive" title="b3">F</font> is   <a href="osalg_4.html#NM4" title="OSALG_4:NM.4">OSCongruence</a> of <font color="Olive" title="b1">U1</font></span><br/><b>let </b><font color="Maroon" title="c2">U1</font>, <font color="Maroon" title="c3">U2</font> be  <a href="msualg_1.html#V5" title="MSUALG_1:attr.5">non-empty</a> <a href="osalg_1.html#NM5" title="OSALG_1:NM.5">OSAlgebra</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"> for <font color="Olive" title="b1">F</font> being  <a href="msualg_3.html#NM1" title="MSUALG_3:NM.1">ManySortedFunction</a> of <font color="Maroon" title="c2">U1</font>,<font color="Maroon" title="c3">U2</font>  st <font color="Olive" title="b1">F</font> <a href="msualg_3.html#R1" title="MSUALG_3:pred.1">is_homomorphism</a> <font color="Maroon" title="c2">U1</font>,<font color="Maroon" title="c3">U2</font> &amp; <font color="Olive" title="b1">F</font> is <a href="osalg_3.html#V1" title="OSALG_3:attr.1">order-sorted</a> holds <br/> <a href="msualg_4.html#K18" title="MSUALG_4:func.18">MSCng</a> <font color="Olive" title="b1">F</font> is   <a href="osalg_4.html#NM4" title="OSALG_4:NM.4">OSCongruence</a> of <font color="Maroon" title="c2">U1</font></span><br/><b>let </b><font color="Maroon" title="c4">F</font> be   <a href="msualg_3.html#NM1" title="MSUALG_3:NM.1">ManySortedFunction</a> of <font color="Maroon" title="c2">U1</font>,<font color="Maroon" title="c3">U2</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="msualg_3.html#R1" title="MSUALG_3:pred.1">is_homomorphism</a> <font color="Maroon" title="c2">U1</font>,<font color="Maroon" title="c3">U2</font> &amp; <font color="Maroon" title="c4">F</font> is <a href="osalg_3.html#V1" title="OSALG_3:attr.1">order-sorted</a> implies  <a href="msualg_4.html#K18" title="MSUALG_4:func.18">MSCng</a> <font color="Maroon" title="c4">F</font> is   <a href="osalg_4.html#NM4" title="OSALG_4:NM.4">OSCongruence</a> of <font color="Maroon" title="c2">U1</font> )</span><br/>





<b>assume </b><a NAME="E1:52"/><i><font color="Green" title="E27">A1</font></i>: 
( <font color="Maroon" title="c4">F</font> <a href="msualg_3.html#R1" title="MSUALG_3:pred.1">is_homomorphism</a> <font color="Maroon" title="c2">U1</font>,<font color="Maroon" title="c3">U2</font> &amp; <font color="Maroon" title="c4">F</font> is <a href="osalg_3.html#V1" title="OSALG_3:attr.1">order-sorted</a> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="msualg_4.html#K18" title="MSUALG_4:func.18">MSCng</a> <font color="Maroon" title="c4">F</font> is   <a href="osalg_4.html#NM4" title="OSALG_4:NM.4">OSCongruence</a> of <font color="Maroon" title="c2">U1</font></span><br/>

<b>reconsider </b><font color="Maroon" title="c5">S1</font> = the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <font color="Maroon" title="c2">U1</font> as   <a href="osalg_1.html#NM4" title="OSALG_1:NM.4">OrderSortedSet</a> of <font color="Maroon" title="c1">S</font> <b>by </b><i><a class="ref" href="osalg_1.html#T17" title="OSALG_1:th.17">OSALG_1:17</a></i>;<br/>
<a NAME="E3:52"/>
 <a href="msualg_4.html#K18" title="MSUALG_4:func.18">MSCng</a> <font color="Maroon" title="c4">F</font> is <a href="osalg_4.html#V1" title="OSALG_4:attr.1">os-compatible</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="52_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c6">s1</font>, <font color="Maroon" title="c7">s2</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">S</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="osalg_4.html#D1" target="_self" title="OSALG_4:def.1">OSALG_4:def 1</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">s1</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c7">s2</font> implies for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <font color="Maroon" title="c2">U1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">s1</font> &amp; <font color="Olive" title="b2">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <font color="Maroon" title="c2">U1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">s1</font> holds <br/>( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="msualg_4.html#K18" title="MSUALG_4:func.18">MSCng</a> <font color="Maroon" title="c4">F</font></span>)</span> <a href="msualg_4.html#K1" title="MSUALG_4:func.1">.</a> <font color="Maroon" title="c6">s1</font> iff <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="msualg_4.html#K18" title="MSUALG_4:func.18">MSCng</a> <font color="Maroon" title="c4">F</font></span>)</span> <a href="msualg_4.html#K1" title="MSUALG_4:func.1">.</a> <font color="Maroon" title="c7">s2</font> ) )</span><br/>

<b>assume </b><a NAME="E1:52_1"/><i><font color="Green" title="E28">A2</font></i>: 
<font color="Maroon" title="c6">s1</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c7">s2</font>
 ; <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">x</font>, <font color="Olive" title="b2">y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <font color="Maroon" title="c2">U1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">s1</font> &amp; <font color="Olive" title="b2">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <font color="Maroon" title="c2">U1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">s1</font> holds <br/>( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="msualg_4.html#K18" title="MSUALG_4:func.18">MSCng</a> <font color="Maroon" title="c4">F</font></span>)</span> <a href="msualg_4.html#K1" title="MSUALG_4:func.1">.</a> <font color="Maroon" title="c6">s1</font> iff <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="msualg_4.html#K18" title="MSUALG_4:func.18">MSCng</a> <font color="Maroon" title="c4">F</font></span>)</span> <a href="msualg_4.html#K1" title="MSUALG_4:func.1">.</a> <font color="Maroon" title="c7">s2</font> )</span><br/>

<b>reconsider </b><font color="Maroon" title="c8">s3</font> = <font color="Maroon" title="c6">s1</font>, <font color="Maroon" title="c9">s4</font> = <font color="Maroon" title="c7">s2</font> as   <a href="osalg_1.html#NM1" title="OSALG_1:NM.1">SortSymbol</a> of <font color="Maroon" title="c1">S</font> ;<br/>
<b>let </b><font color="Maroon" title="c10">x</font>, <font color="Maroon" title="c11">y</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="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <font color="Maroon" title="c2">U1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">s1</font> &amp; <font color="Maroon" title="c11">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <font color="Maroon" title="c2">U1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">s1</font> implies ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c10">x</font>,<font color="Maroon" title="c11">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="msualg_4.html#K18" title="MSUALG_4:func.18">MSCng</a> <font color="Maroon" title="c4">F</font></span>)</span> <a href="msualg_4.html#K1" title="MSUALG_4:func.1">.</a> <font color="Maroon" title="c6">s1</font> iff <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c10">x</font>,<font color="Maroon" title="c11">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="msualg_4.html#K18" title="MSUALG_4:func.18">MSCng</a> <font color="Maroon" title="c4">F</font></span>)</span> <a href="msualg_4.html#K1" title="MSUALG_4:func.1">.</a> <font color="Maroon" title="c7">s2</font> ) )</span><br/>

<b>assume </b><a NAME="E3:52_1"/><i><font color="Green" title="E29">A3</font></i>: 
( <font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <font color="Maroon" title="c2">U1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">s1</font> &amp; <font color="Maroon" title="c11">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <font color="Maroon" title="c2">U1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">s1</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c10">x</font>,<font color="Maroon" title="c11">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="msualg_4.html#K18" title="MSUALG_4:func.18">MSCng</a> <font color="Maroon" title="c4">F</font></span>)</span> <a href="msualg_4.html#K1" title="MSUALG_4:func.1">.</a> <font color="Maroon" title="c6">s1</font> iff <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c10">x</font>,<font color="Maroon" title="c11">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="msualg_4.html#K18" title="MSUALG_4:func.18">MSCng</a> <font color="Maroon" title="c4">F</font></span>)</span> <a href="msualg_4.html#K1" title="MSUALG_4:func.1">.</a> <font color="Maroon" title="c7">s2</font> )</span><br/>

<a NAME="E4:52_1"/><i><font color="Green" title="E30">A4</font></i>: 
<font color="Maroon" title="c5">S1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">s3</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">S1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">s4</font>
 
<b>by </b><i><a class="txt" href="osalg_4.html#E1:52_1"><i><font color="Green" title="E28">A2</font></i></a>, <a class="ref" href="osalg_1.html#D18" title="OSALG_1:def.18">OSALG_1:def 18</a></i>;<br/>
<a NAME="E5:52_1"/><i><font color="Green" title="E31">A5</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="c4">F</font> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <font color="Maroon" title="c8">s3</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">S1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">s3</font> &amp;  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">F</font> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <font color="Maroon" title="c9">s4</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">S1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">s4</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="c12">x1</font> = <font color="Maroon" title="c10">x</font>, <font color="Maroon" title="c13">y1</font> = <font color="Maroon" title="c11">y</font> 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 <font color="Maroon" title="c2">U1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">s1</font> <b>by </b><i><a class="txt" href="osalg_4.html#E3:52_1"><i><font color="Green" title="E29">A3</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c14">x2</font> = <font color="Maroon" title="c10">x</font>, <font color="Maroon" title="c15">y2</font> = <font color="Maroon" title="c11">y</font> 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 <font color="Maroon" title="c2">U1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">s2</font> <b>by </b><i><a class="txt" href="osalg_4.html#E3:52_1"><i><font color="Green" title="E29">A3</font></i></a>, <a class="txt" href="osalg_4.html#E4:52_1"><i><font color="Green" title="E30">A4</font></i></a></i>;<br/>
<a NAME="E8:52_1"/><i><font color="Green" title="E32">A6</font></i>: 
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c14">x2</font>,<font color="Maroon" title="c15">y2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="msualg_4.html#K17" title="MSUALG_4:func.17">MSCng</a> <font color="Maroon" title="c4">F</font>,<font color="Maroon" title="c7">s2</font> iff <span class="p1">(<span class="default"><font color="Maroon" title="c4">F</font> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <font color="Maroon" title="c7">s2</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c14">x2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">F</font> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <font color="Maroon" title="c7">s2</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c15">y2</font> )
 
<b>by </b><i><a class="ref" href="msualg_4.html#D19" title="MSUALG_4:def.19">MSUALG_4:def 19</a></i>;<br/>
<a NAME="E9:52_1"/><i><font color="Green" title="E33">A7</font></i>: 
( <font color="Maroon" title="c12">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">F</font> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <font color="Maroon" title="c9">s4</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c4">F</font> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <font color="Maroon" title="c8">s3</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">F</font> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <font color="Maroon" title="c9">s4</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">x1</font> &amp; <font color="Maroon" title="c13">y1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">F</font> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <font color="Maroon" title="c9">s4</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c4">F</font> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <font color="Maroon" title="c8">s3</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">F</font> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <font color="Maroon" title="c9">s4</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">y1</font> )
 
<b>by </b><i><a class="txt" href="osalg_4.html#E1:52"><i><font color="Green" title="E27">A1</font></i></a>, <a class="txt" href="osalg_4.html#E1:52_1"><i><font color="Green" title="E28">A2</font></i></a>, <a class="txt" href="osalg_4.html#E5:52_1"><i><font color="Green" title="E31">A5</font></i></a>, <a class="ref" href="osalg_3.html#D1" title="OSALG_3:def.1">OSALG_3:def 1</a></i>;<br/>
<a NAME="E10:52_1"/>
( <span class="p1">(<span class="default"><a href="msualg_4.html#K18" title="MSUALG_4:func.18">MSCng</a> <font color="Maroon" title="c4">F</font></span>)</span> <a href="msualg_4.html#K2" title="MSUALG_4:func.2">.</a> <font color="Maroon" title="c6">s1</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a>  <a href="msualg_4.html#K17" title="MSUALG_4:func.17">MSCng</a> <font color="Maroon" title="c4">F</font>,<font color="Maroon" title="c6">s1</font> &amp; <span class="p1">(<span class="default"><a href="msualg_4.html#K18" title="MSUALG_4:func.18">MSCng</a> <font color="Maroon" title="c4">F</font></span>)</span> <a href="msualg_4.html#K2" title="MSUALG_4:func.2">.</a> <font color="Maroon" title="c7">s2</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a>  <a href="msualg_4.html#K17" title="MSUALG_4:func.17">MSCng</a> <font color="Maroon" title="c4">F</font>,<font color="Maroon" title="c7">s2</font> )
 
<b>by </b><i><a class="txt" href="osalg_4.html#E1:52"><i><font color="Green" title="E27">A1</font></i></a>, <a class="ref" href="msualg_4.html#D20" title="MSUALG_4:def.20">MSUALG_4:def 20</a></i>;<br/>
<b>hence </b><a NAME="E11:52_1"/>
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c10">x</font>,<font color="Maroon" title="c11">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="msualg_4.html#K18" title="MSUALG_4:func.18">MSCng</a> <font color="Maroon" title="c4">F</font></span>)</span> <a href="msualg_4.html#K2" title="MSUALG_4:func.2">.</a> <font color="Maroon" title="c6">s1</font> iff <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c10">x</font>,<font color="Maroon" title="c11">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="msualg_4.html#K18" title="MSUALG_4:func.18">MSCng</a> <font color="Maroon" title="c4">F</font></span>)</span> <a href="msualg_4.html#K2" title="MSUALG_4:func.2">.</a> <font color="Maroon" title="c7">s2</font> )
 <b>by </b><i><a class="txt" href="osalg_4.html#E8:52_1"><i><font color="Green" title="E32">A6</font></i></a>, <a class="txt" href="osalg_4.html#E9:52_1"><i><font color="Green" title="E33">A7</font></i></a>, <a class="ref" href="msualg_4.html#D19" title="MSUALG_4:def.19">MSUALG_4:def 19</a></i>; <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="E4:52"/>
 <a href="msualg_4.html#K18" title="MSUALG_4:func.18">MSCng</a> <font color="Maroon" title="c4">F</font> is   <a href="osalg_4.html#NM4" title="OSALG_4:NM.4">OSCongruence</a> of <font color="Maroon" title="c2">U1</font>
 <b>by </b><i><a class="ref" href="osalg_4.html#D3" target="_self" title="OSALG_4:def.3">Def3</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div>
