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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" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  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> of <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#NM5" title="MSUALG_4:NM.5">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> of <font color="Maroon" title="c1">S</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  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#NM5" title="MSUALG_4:NM.5">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" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  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#NM5" title="MSUALG_4:NM.5">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" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">R</font> being  <a href="msualg_4.html#NM5" title="MSUALG_4:NM.5">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#NM5" title="MSUALG_4:NM.5">MSCongruence</a> of <font color="Maroon" title="c2">U0</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  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="E1:53"/><span class="lab"><font color="Green" title="E25">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" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  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="Red">:: according to </font></span><a class="ref" href="equation.html#D5" target="_self" title="EQUATION:def.5">EQUATION:def 5</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  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="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="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="mcart_1.html#K2" title="MCART_1:func.2">`2</a> </span>)</span> )</span><br/>

<span class="kw">assume </span><a NAME="E2:53"/>
<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" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  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="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="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="mcart_1.html#K2" title="MCART_1: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="E4:53"/><span class="lab"><font color="Green" title="E26">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="E5:53"/><span class="lab"><font color="Green" title="E27">A3</font></span>: 
<font color="Maroon" title="c7">h</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</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#T26" target="_self" title="EQUATION:th.26">Th26</a>, <a class="ref" href="msualg_4.html#T3" title="MSUALG_4:th.3">MSUALG_4:3</a></span>;<br/>
<a NAME="E6:53"/><span class="lab"><font color="Green" title="E28">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" title="FUNCT_2:def.1">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="mcart_1.html#K1" title="MCART_1: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="mcart_1.html#K1" title="MCART_1:func.1">`1</a> </span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:53"><span class="lab"><font color="Green" title="E27">A3</font></span></a>, <a class="ref" href="msualg_3.html#T2" title="MSUALG_3:th.2">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="mcart_1.html#K1" title="MCART_1:func.1">`1</a> </span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:53"><span class="lab"><font color="Green" title="E28">A4</font></span></a>, <a class="ref" href="equation.html#T31" target="_self" title="EQUATION:th.31">Th31</a>, <a class="ref" href="funct_1.html#T23" title="FUNCT_1:th.23">FUNCT_1:23</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="mcart_1.html#K2" title="MCART_1:func.2">`2</a> </span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:53"><span class="lab"><font color="Green" title="E25">A1</font></span></a>, <a class="txt" href="#E4:53"><span class="lab"><font color="Green" title="E26">A2</font></span></a>, <a class="ref" href="equation.html#D5" target="_self" title="EQUATION:def.5">Def5</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="mcart_1.html#K2" title="MCART_1:func.2">`2</a> </span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:53"><span class="lab"><font color="Green" title="E28">A4</font></span></a>, <a class="ref" href="equation.html#T32" target="_self" title="EQUATION:th.32">Th32</a>, <a class="ref" href="funct_1.html#T23" title="FUNCT_1:th.23">FUNCT_1:23</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="mcart_1.html#K2" title="MCART_1:func.2">`2</a> </span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:53"><span class="lab"><font color="Green" title="E27">A3</font></span></a>, <a class="ref" href="msualg_3.html#T2" title="MSUALG_3:th.2">MSUALG_3:2</a></span>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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