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<span class="kw">let </span><font color="Maroon" title="c1">S</font> be   non  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>   non  <a href="struct_0.html#V11" title="STRUCT_0:attr.11">void</a>   <a href="msualg_1.html#L1" title="MSUALG_1:struct.1">ManySortedSign</a> ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">U0</font> being   <a href="msualg_1.html#V4" title="MSUALG_1:attr.4">non-empty</a>   <a href="msualg_1.html#L3" title="MSUALG_1:struct.3">MSAlgebra</a> over <font color="Maroon" title="c1">S</font><br/>  for <font color="Olive" title="b2">s</font> being   <a href="msualg_1.html#NM1" title="MSUALG_1:NM.1">SortSymbol</a> of <font color="Maroon" title="c1">S</font><br/>  for <font color="Olive" title="b3">e</font> being    <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="Olive" title="b2">s</font><br/>  for <font color="Olive" title="b4">R</font> being   <a href="msualg_4.html#NM4" title="MSUALG_4:NM.4">MSCongruence</a> of <font color="Olive" title="b1">U0</font>  st <font color="Olive" title="b1">U0</font> <a href="equation.html#R1" title="EQUATION:pred.1">|=</a> <font color="Olive" title="b3">e</font> holds <br/> <a href="msualg_4.html#K13" title="MSUALG_4:func.13">QuotMSAlg</a> (<font color="Olive" title="b1">U0</font>,<font color="Olive" title="b4">R</font>) <a href="equation.html#R1" title="EQUATION:pred.1">|=</a> <font color="Olive" title="b3">e</font></span><br/><span class="kw">let </span><font color="Maroon" title="c2">U0</font> be   <a href="msualg_1.html#V4" title="MSUALG_1:attr.4">non-empty</a>   <a href="msualg_1.html#L3" title="MSUALG_1:struct.3">MSAlgebra</a> over <font color="Maroon" title="c1">S</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">s</font> being   <a href="msualg_1.html#NM1" title="MSUALG_1:NM.1">SortSymbol</a> of <font color="Maroon" title="c1">S</font><br/>  for <font color="Olive" title="b2">e</font> being    <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="Olive" title="b1">s</font><br/>  for <font color="Olive" title="b3">R</font> being   <a href="msualg_4.html#NM4" title="MSUALG_4:NM.4">MSCongruence</a> of <font color="Maroon" title="c2">U0</font>  st <font color="Maroon" title="c2">U0</font> <a href="equation.html#R1" title="EQUATION:pred.1">|=</a> <font color="Olive" title="b2">e</font> holds <br/> <a href="msualg_4.html#K13" title="MSUALG_4:func.13">QuotMSAlg</a> (<font color="Maroon" title="c2">U0</font>,<font color="Olive" title="b3">R</font>) <a href="equation.html#R1" title="EQUATION:pred.1">|=</a> <font color="Olive" title="b2">e</font></span><br/><span class="kw">let </span><font color="Maroon" title="c3">s</font> be   <a href="msualg_1.html#NM1" title="MSUALG_1:NM.1">SortSymbol</a> of <font color="Maroon" title="c1">S</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">e</font> being    <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="c3">s</font><br/>  for <font color="Olive" title="b2">R</font> being   <a href="msualg_4.html#NM4" title="MSUALG_4:NM.4">MSCongruence</a> of <font color="Maroon" title="c2">U0</font>  st <font color="Maroon" title="c2">U0</font> <a href="equation.html#R1" title="EQUATION:pred.1">|=</a> <font color="Olive" title="b1">e</font> holds <br/> <a href="msualg_4.html#K13" title="MSUALG_4:func.13">QuotMSAlg</a> (<font color="Maroon" title="c2">U0</font>,<font color="Olive" title="b2">R</font>) <a href="equation.html#R1" title="EQUATION:pred.1">|=</a> <font color="Olive" title="b1">e</font></span><br/><span class="kw">let </span><font color="Maroon" title="c4">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="c3">s</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">R</font> being   <a href="msualg_4.html#NM4" title="MSUALG_4:NM.4">MSCongruence</a> of <font color="Maroon" title="c2">U0</font>  st <font color="Maroon" title="c2">U0</font> <a href="equation.html#R1" title="EQUATION:pred.1">|=</a> <font color="Maroon" title="c4">e</font> holds <br/> <a href="msualg_4.html#K13" title="MSUALG_4:func.13">QuotMSAlg</a> (<font color="Maroon" title="c2">U0</font>,<font color="Olive" title="b1">R</font>) <a href="equation.html#R1" title="EQUATION:pred.1">|=</a> <font color="Maroon" title="c4">e</font></span><br/><span class="kw">let </span><font color="Maroon" title="c5">R</font> be   <a href="msualg_4.html#NM4" title="MSUALG_4:NM.4">MSCongruence</a> of <font color="Maroon" title="c2">U0</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c2">U0</font> <a href="equation.html#R1" title="EQUATION:pred.1">|=</a> <font color="Maroon" title="c4">e</font> implies  <a href="msualg_4.html#K13" title="MSUALG_4:func.13">QuotMSAlg</a> (<font color="Maroon" title="c2">U0</font>,<font color="Maroon" title="c5">R</font>) <a href="equation.html#R1" title="EQUATION:pred.1">|=</a> <font color="Maroon" title="c4">e</font> )</span><br/>









<span class="kw">assume </span><a NAME="E1:48"/><span class="lab"><font color="Green" title="E23">A1</font></span>: 
<font color="Maroon" title="c2">U0</font> <a href="equation.html#R1" title="EQUATION:pred.1">|=</a> <font color="Maroon" title="c4">e</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="msualg_4.html#K13" title="MSUALG_4:func.13">QuotMSAlg</a> (<font color="Maroon" title="c2">U0</font>,<font color="Maroon" title="c5">R</font>) <a href="equation.html#R1" title="EQUATION:pred.1">|=</a> <font color="Maroon" title="c4">e</font></span><br/>

<span class="kw">set </span><font color="Maroon" title="c6">n</font> = <span class="p1">(<span class="default"><a href="msualg_4.html#K15" title="MSUALG_4:func.15">MSNat_Hom</a> (<font color="Maroon" title="c2">U0</font>,<font color="Maroon" title="c5">R</font>)</span>)</span> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <font color="Maroon" title="c3">s</font>;<br/>
<span class="kw">let </span><font color="Maroon" title="c7">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="msualg_4.html#K13" title="MSUALG_4:func.13">QuotMSAlg</a> (<font color="Maroon" title="c2">U0</font>,<font color="Maroon" title="c5">R</font>)</span>)</span>; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="equation.html#D5" target="_self" onmouseover="rs('equation/D5')" onmouseout="rh()">EQUATION:def 5</a> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c7">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="msualg_4.html#K13" title="MSUALG_4:func.13">QuotMSAlg</a> (<font color="Maroon" title="c2">U0</font>,<font color="Maroon" title="c5">R</font>) implies <span class="p1">(<span class="default"><font color="Maroon" title="c7">h</font> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <font color="Maroon" title="c3">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="c4">e</font> <a href="xtuple_0.html#K1" title="XTUPLE_0: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="c7">h</font> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <font color="Maroon" title="c3">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="c4">e</font> <a href="xtuple_0.html#K2" title="XTUPLE_0:func.2">`2</a></span>)</span> )</span><br/>

<span class="kw">assume </span><a NAME="E2:48"/>
<font color="Maroon" title="c7">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="msualg_4.html#K13" title="MSUALG_4:func.13">QuotMSAlg</a> (<font color="Maroon" title="c2">U0</font>,<font color="Maroon" title="c5">R</font>)
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c7">h</font> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <font color="Maroon" title="c3">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="c4">e</font> <a href="xtuple_0.html#K1" title="XTUPLE_0: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="c7">h</font> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <font color="Maroon" title="c3">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="c4">e</font> <a href="xtuple_0.html#K2" title="XTUPLE_0:func.2">`2</a></span>)</span></span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c8">h0</font> being   <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>,<font color="Maroon" title="c2">U0</font><span class="kw"> such that </span><br/><a NAME="E4:48"/><span class="lab"><font color="Green" title="E24">A2</font></span>: 
<font color="Maroon" title="c8">h0</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="c2">U0</font>
 <span class="kw">and </span><br/><a NAME="E5:48"/><span class="lab"><font color="Green" title="E25">A3</font></span>: 
<font color="Maroon" title="c7">h</font> <a href="pboole.html#R8" title="PBOOLE:pred.8">=</a> <span class="p1">(<span class="default"><a href="msualg_4.html#K15" title="MSUALG_4:func.15">MSNat_Hom</a> (<font color="Maroon" title="c2">U0</font>,<font color="Maroon" title="c5">R</font>)</span>)</span> <a href="msualg_3.html#K3" title="MSUALG_3:func.3">**</a> <font color="Maroon" title="c8">h0</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="equation.html#T24" target="_self" onmouseover="rs('equation/T24')" onmouseout="rh()">Th24</a>, <a class="ref" href="msualg_4.html#T3" onmouseover="rs('msualg_4/T3')" onmouseout="rh()">MSUALG_4:3</a></span>;<br/>
<a NAME="E6:48"/><span class="lab"><font color="Green" title="E26">A4</font></span>: 
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">h0</font> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <font color="Maroon" title="c3">s</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">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="c3">s</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D1" onmouseover="rs('funct_2/D1')" onmouseout="rh()">FUNCT_2:def 1</a></span>;<br/>
<span class="kw">thus </span><span class="p1">(<span class="default"><font color="Maroon" title="c7">h</font> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <font color="Maroon" title="c3">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="c4">e</font> <a href="xtuple_0.html#K1" title="XTUPLE_0:func.1">`1</a></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="msualg_4.html#K15" title="MSUALG_4:func.15">MSNat_Hom</a> (<font color="Maroon" title="c2">U0</font>,<font color="Maroon" title="c5">R</font>)</span>)</span> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <font color="Maroon" title="c3">s</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">h0</font> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <font color="Maroon" title="c3">s</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">e</font> <a href="xtuple_0.html#K1" title="XTUPLE_0:func.1">`1</a></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:48"><span class="lab"><font color="Green" title="E25">A3</font></span></a>, <a class="ref" href="msualg_3.html#T2" onmouseover="rs('msualg_3/T2')" onmouseout="rh()">MSUALG_3:2</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="msualg_4.html#K15" title="MSUALG_4:func.15">MSNat_Hom</a> (<font color="Maroon" title="c2">U0</font>,<font color="Maroon" title="c5">R</font>)</span>)</span> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <font color="Maroon" title="c3">s</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="c8">h0</font> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <font color="Maroon" title="c3">s</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">e</font> <a href="xtuple_0.html#K1" title="XTUPLE_0:func.1">`1</a></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:48"><span class="lab"><font color="Green" title="E26">A4</font></span></a>, <a class="ref" href="equation.html#T29" target="_self" onmouseover="rs('equation/T29')" onmouseout="rh()">Th29</a>, <a class="ref" href="funct_1.html#T13" onmouseover="rs('funct_1/T13')" onmouseout="rh()">FUNCT_1:13</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="msualg_4.html#K15" title="MSUALG_4:func.15">MSNat_Hom</a> (<font color="Maroon" title="c2">U0</font>,<font color="Maroon" title="c5">R</font>)</span>)</span> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <font color="Maroon" title="c3">s</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="c8">h0</font> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <font color="Maroon" title="c3">s</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">e</font> <a href="xtuple_0.html#K2" title="XTUPLE_0:func.2">`2</a></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:48"><span class="lab"><font color="Green" title="E23">A1</font></span></a>, <a class="txt" href="#E4:48"><span class="lab"><font color="Green" title="E24">A2</font></span></a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="msualg_4.html#K15" title="MSUALG_4:func.15">MSNat_Hom</a> (<font color="Maroon" title="c2">U0</font>,<font color="Maroon" title="c5">R</font>)</span>)</span> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <font color="Maroon" title="c3">s</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">h0</font> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <font color="Maroon" title="c3">s</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">e</font> <a href="xtuple_0.html#K2" title="XTUPLE_0:func.2">`2</a></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:48"><span class="lab"><font color="Green" title="E26">A4</font></span></a>, <a class="ref" href="equation.html#T30" target="_self" onmouseover="rs('equation/T30')" onmouseout="rh()">Th30</a>, <a class="ref" href="funct_1.html#T13" onmouseover="rs('funct_1/T13')" onmouseout="rh()">FUNCT_1:13</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c7">h</font> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <font color="Maroon" title="c3">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="c4">e</font> <a href="xtuple_0.html#K2" title="XTUPLE_0:func.2">`2</a></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:48"><span class="lab"><font color="Green" title="E25">A3</font></span></a>, <a class="ref" href="msualg_3.html#T2" onmouseover="rs('msualg_3/T2')" onmouseout="rh()">MSUALG_3:2</a></span>
; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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