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<b>let </b><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:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">U0</font>, <font color="Olive" title="b2">U1</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="b3">E</font> being  <a href="equation.html#NM1" title="EQUATION:NM.1">EqualSet</a> of <font color="Maroon" title="c1">S</font>  st <font color="Olive" title="b1">U0</font>,<font color="Olive" title="b2">U1</font> <a href="msualg_3.html#R6" title="MSUALG_3:pred.6">are_isomorphic</a>  &amp; <font color="Olive" title="b1">U0</font> <a href="equation.html#R2" title="EQUATION:pred.2">|=</a> <font color="Olive" title="b3">E</font> holds <br/><font color="Olive" title="b2">U1</font> <a href="equation.html#R2" title="EQUATION:pred.2">|=</a> <font color="Olive" title="b3">E</font></span><br/><b>let </b><font color="Maroon" title="c2">U0</font>, <font color="Maroon" title="c3">U1</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:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">E</font> being  <a href="equation.html#NM1" title="EQUATION:NM.1">EqualSet</a> of <font color="Maroon" title="c1">S</font>  st <font color="Maroon" title="c2">U0</font>,<font color="Maroon" title="c3">U1</font> <a href="msualg_3.html#R6" title="MSUALG_3:pred.6">are_isomorphic</a>  &amp; <font color="Maroon" title="c2">U0</font> <a href="equation.html#R2" title="EQUATION:pred.2">|=</a> <font color="Olive" title="b1">E</font> holds <br/><font color="Maroon" title="c3">U1</font> <a href="equation.html#R2" title="EQUATION:pred.2">|=</a> <font color="Olive" title="b1">E</font></span><br/><b>let </b><font color="Maroon" title="c4">E</font> be   <a href="equation.html#NM1" title="EQUATION:NM.1">EqualSet</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="c2">U0</font>,<font color="Maroon" title="c3">U1</font> <a href="msualg_3.html#R6" title="MSUALG_3:pred.6">are_isomorphic</a>  &amp; <font color="Maroon" title="c2">U0</font> <a href="equation.html#R2" title="EQUATION:pred.2">|=</a> <font color="Maroon" title="c4">E</font> implies <font color="Maroon" title="c3">U1</font> <a href="equation.html#R2" title="EQUATION:pred.2">|=</a> <font color="Maroon" title="c4">E</font> )</span><br/>







<b>assume </b><b>that </b><br/><a NAME="E1:50"/><i><font color="Green" title="E24">A1</font></i>: 
<font color="Maroon" title="c2">U0</font>,<font color="Maroon" title="c3">U1</font> <a href="msualg_3.html#R6" title="MSUALG_3:pred.6">are_isomorphic</a> 
 <b>and </b><br/><a NAME="E2:50"/><i><font color="Green" title="E25">A2</font></i>: 
<font color="Maroon" title="c2">U0</font> <a href="equation.html#R2" title="EQUATION:pred.2">|=</a> <font color="Maroon" title="c4">E</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c3">U1</font> <a href="equation.html#R2" title="EQUATION:pred.2">|=</a> <font color="Maroon" title="c4">E</font></span><br/>

<b>let </b><font color="Maroon" title="c5">s</font> be   <a href="msualg_1.html#NM1" title="MSUALG_1:NM.1">SortSymbol</a> of <font color="Maroon" title="c1">S</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="equation.html#D6" target="_self" title="EQUATION:def.6">EQUATION:def 6</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">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="c5">s</font>  st <font color="Olive" title="b1">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">E</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">s</font> holds <br/><font color="Maroon" title="c3">U1</font> <a href="equation.html#R1" title="EQUATION:pred.1">|=</a> <font color="Olive" title="b1">e</font></span><br/><b>let </b><font color="Maroon" title="c6">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="c5">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="c6">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">E</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">s</font> implies <font color="Maroon" title="c3">U1</font> <a href="equation.html#R1" title="EQUATION:pred.1">|=</a> <font color="Maroon" title="c6">e</font> )</span><br/>



<b>assume </b><a NAME="E3:50"/>
<font color="Maroon" title="c6">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">E</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">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="c3">U1</font> <a href="equation.html#R1" title="EQUATION:pred.1">|=</a> <font color="Maroon" title="c6">e</font></span><br/>

<a NAME="E4:50"/><b>then </b>
<font color="Maroon" title="c2">U0</font> <a href="equation.html#R1" title="EQUATION:pred.1">|=</a> <font color="Maroon" title="c6">e</font>
 
<b>by </b><i><a class="txt" href="#E2:50"><i><font color="Green" title="E25">A2</font></i></a>, <a class="ref" href="equation.html#D6" target="_self" title="EQUATION:def.6">Def6</a></i>;<br/>
<b>hence </b><a NAME="E5:50"/>
<font color="Maroon" title="c3">U1</font> <a href="equation.html#R1" title="EQUATION:pred.1">|=</a> <font color="Maroon" title="c6">e</font>
 <b>by </b><i><a class="txt" href="#E1:50"><i><font color="Green" title="E24">A1</font></i></a>, <a class="ref" href="equation.html#T35" target="_self" title="EQUATION:th.35">Th35</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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