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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="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">s1</font>, <font color="Olive" title="b2">s2</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">A</font> being    <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="b4">f</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a>  st <font color="Olive" title="b4">f</font> <a href="msualg_6.html#R1" title="MSUALG_6:pred.1">is_e.translation_of</a> <font color="Olive" title="b3">A</font>,<font color="Olive" title="b1">s1</font>,<font color="Olive" title="b2">s2</font> holds <br/><span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">s1</font>,<font color="Olive" title="b2">s2</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_6.html#K3" title="MSUALG_6:func.3">TranslationRel</a> <font color="Maroon" title="c1">S</font></span><br/><span class="kw">let </span><font color="Maroon" title="c2">s1</font>, <font color="Maroon" title="c3">s2</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="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">A</font> being    <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">f</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a>  st <font color="Olive" title="b2">f</font> <a href="msualg_6.html#R1" title="MSUALG_6:pred.1">is_e.translation_of</a> <font color="Olive" title="b1">A</font>,<font color="Maroon" title="c2">s1</font>,<font color="Maroon" title="c3">s2</font> holds <br/><span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c2">s1</font>,<font color="Maroon" title="c3">s2</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_6.html#K3" title="MSUALG_6:func.3">TranslationRel</a> <font color="Maroon" title="c1">S</font></span><br/><span class="kw">let </span><font color="Maroon" title="c4">A</font> be    <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="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">f</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a>  st <font color="Olive" title="b1">f</font> <a href="msualg_6.html#R1" title="MSUALG_6:pred.1">is_e.translation_of</a> <font color="Maroon" title="c4">A</font>,<font color="Maroon" title="c2">s1</font>,<font color="Maroon" title="c3">s2</font> holds <br/><span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c2">s1</font>,<font color="Maroon" title="c3">s2</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_6.html#K3" title="MSUALG_6:func.3">TranslationRel</a> <font color="Maroon" title="c1">S</font></span><br/><span class="kw">let </span><font color="Maroon" title="c5">f</font> be   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c5">f</font> <a href="msualg_6.html#R1" title="MSUALG_6:pred.1">is_e.translation_of</a> <font color="Maroon" title="c4">A</font>,<font color="Maroon" title="c2">s1</font>,<font color="Maroon" title="c3">s2</font> implies <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c2">s1</font>,<font color="Maroon" title="c3">s2</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_6.html#K3" title="MSUALG_6:func.3">TranslationRel</a> <font color="Maroon" title="c1">S</font> )</span><br/>







<span class="kw">assume </span><a NAME="E1:22"/>
<font color="Maroon" title="c5">f</font> <a href="msualg_6.html#R1" title="MSUALG_6:pred.1">is_e.translation_of</a> <font color="Maroon" title="c4">A</font>,<font color="Maroon" title="c2">s1</font>,<font color="Maroon" title="c3">s2</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c2">s1</font>,<font color="Maroon" title="c3">s2</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_6.html#K3" title="MSUALG_6:func.3">TranslationRel</a> <font color="Maroon" title="c1">S</font></span><br/>

<a NAME="E2:22"/><span class="kw">then </span>
 ex <font color="Olive" title="b1">o</font> being   <a href="msualg_1.html#NM2" title="MSUALG_1:NM.2">OperSymbol</a> of <font color="Maroon" title="c1">S</font> st <br/>(  <a href="msualg_1.html#K2" title="MSUALG_1:func.2">the_result_sort_of</a> <font color="Olive" title="b1">o</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">s2</font> &amp;  ex <font color="Olive" title="b2">i</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( <font color="Olive" title="b2">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K1" title="MSUALG_1:func.1">the_arity_of</a> <font color="Olive" title="b1">o</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="msualg_1.html#K1" title="MSUALG_1:func.1">the_arity_of</a> <font color="Olive" title="b1">o</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b2">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">s1</font> &amp;  ex <font color="Olive" title="b3">a</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a> st <br/>( <font color="Olive" title="b3">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="msualg_1.html#K3" title="MSUALG_1:func.3">Args</a> (<font color="Olive" title="b1">o</font>,<font color="Maroon" title="c4">A</font>) &amp; <font color="Maroon" title="c5">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="msualg_6.html#K4" title="MSUALG_6:func.4">transl</a> (<font color="Olive" title="b1">o</font>,<font color="Olive" title="b2">i</font>,<font color="Olive" title="b3">a</font>,<font color="Maroon" title="c4">A</font>) ) ) )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="msualg_6.html#D5" target="_self" title="MSUALG_6:def.5">Def5</a></span>;<br/>
<span class="kw">hence </span><a NAME="E3:22"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c2">s1</font>,<font color="Maroon" title="c3">s2</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_6.html#K3" title="MSUALG_6:func.3">TranslationRel</a> <font color="Maroon" title="c1">S</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="msualg_6.html#D3" target="_self" title="MSUALG_6:def.3">Def3</a></span>; <a class="txt" 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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