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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">A1</font>, <font color="Olive" title="b2">A2</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">h</font> being  <a href="msualg_3.html#NM1" title="MSUALG_3:NM.1">ManySortedFunction</a> of <font color="Olive" title="b1">A1</font>,<font color="Olive" title="b2">A2</font>  st <font color="Olive" title="b3">h</font> <a href="msualg_3.html#R1" title="MSUALG_3:pred.1">is_homomorphism</a> <font color="Olive" title="b1">A1</font>,<font color="Olive" title="b2">A2</font> holds <br/>for <font color="Olive" title="b4">s1</font>, <font color="Olive" title="b5">s2</font> being  <a href="msualg_1.html#NM1" title="MSUALG_1:NM.1">SortSymbol</a> of <font color="Maroon" title="c1">S</font>  st  <a href="msualg_6.html#K3" title="MSUALG_6:func.3">TranslationRel</a> <font color="Maroon" title="c1">S</font> <a href="rewrite1.html#R1" title="REWRITE1:pred.1">reduces</a> <font color="Olive" title="b4">s1</font>,<font color="Olive" title="b5">s2</font> holds <br/>for <font color="Olive" title="b6">t</font> being   <a href="msualg_6.html#M2" title="MSUALG_6:mode.2">Translation</a> of <font color="Olive" title="b1">A1</font>,<font color="Olive" title="b4">s1</font>,<font color="Olive" title="b5">s2</font>  ex <font color="Olive" title="b7">T</font> being   <a href="msualg_6.html#M2" title="MSUALG_6:mode.2">Translation</a> of <font color="Olive" title="b2">A2</font>,<font color="Olive" title="b4">s1</font>,<font color="Olive" title="b5">s2</font> st <font color="Olive" title="b7">T</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p1">(<span class="default"><font color="Olive" title="b3">h</font> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <font color="Olive" title="b4">s1</font></span>)</span> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b3">h</font> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <font color="Olive" title="b5">s2</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Olive" title="b6">t</font></span><br/><span class="kw">let </span><font color="Maroon" title="c2">A1</font>, <font color="Maroon" title="c3">A2</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">h</font> being  <a href="msualg_3.html#NM1" title="MSUALG_3:NM.1">ManySortedFunction</a> of <font color="Maroon" title="c2">A1</font>,<font color="Maroon" title="c3">A2</font>  st <font color="Olive" title="b1">h</font> <a href="msualg_3.html#R1" title="MSUALG_3:pred.1">is_homomorphism</a> <font color="Maroon" title="c2">A1</font>,<font color="Maroon" title="c3">A2</font> holds <br/>for <font color="Olive" title="b2">s1</font>, <font color="Olive" title="b3">s2</font> being  <a href="msualg_1.html#NM1" title="MSUALG_1:NM.1">SortSymbol</a> of <font color="Maroon" title="c1">S</font>  st  <a href="msualg_6.html#K3" title="MSUALG_6:func.3">TranslationRel</a> <font color="Maroon" title="c1">S</font> <a href="rewrite1.html#R1" title="REWRITE1:pred.1">reduces</a> <font color="Olive" title="b2">s1</font>,<font color="Olive" title="b3">s2</font> holds <br/>for <font color="Olive" title="b4">t</font> being   <a href="msualg_6.html#M2" title="MSUALG_6:mode.2">Translation</a> of <font color="Maroon" title="c2">A1</font>,<font color="Olive" title="b2">s1</font>,<font color="Olive" title="b3">s2</font>  ex <font color="Olive" title="b5">T</font> being   <a href="msualg_6.html#M2" title="MSUALG_6:mode.2">Translation</a> of <font color="Maroon" title="c3">A2</font>,<font color="Olive" title="b2">s1</font>,<font color="Olive" title="b3">s2</font> st <font color="Olive" title="b5">T</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">h</font> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <font color="Olive" title="b2">s1</font></span>)</span> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">h</font> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <font color="Olive" title="b3">s2</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Olive" title="b4">t</font></span><br/><span class="kw">let </span><font color="Maroon" title="c4">h</font> be   <a href="msualg_3.html#NM1" title="MSUALG_3:NM.1">ManySortedFunction</a> of <font color="Maroon" title="c2">A1</font>,<font color="Maroon" title="c3">A2</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="c4">h</font> <a href="msualg_3.html#R1" title="MSUALG_3:pred.1">is_homomorphism</a> <font color="Maroon" title="c2">A1</font>,<font color="Maroon" title="c3">A2</font> implies 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>  st  <a href="msualg_6.html#K3" title="MSUALG_6:func.3">TranslationRel</a> <font color="Maroon" title="c1">S</font> <a href="rewrite1.html#R1" title="REWRITE1:pred.1">reduces</a> <font color="Olive" title="b1">s1</font>,<font color="Olive" title="b2">s2</font> holds <br/>for <font color="Olive" title="b3">t</font> being   <a href="msualg_6.html#M2" title="MSUALG_6:mode.2">Translation</a> of <font color="Maroon" title="c2">A1</font>,<font color="Olive" title="b1">s1</font>,<font color="Olive" title="b2">s2</font>  ex <font color="Olive" title="b4">T</font> being   <a href="msualg_6.html#M2" title="MSUALG_6:mode.2">Translation</a> of <font color="Maroon" title="c3">A2</font>,<font color="Olive" title="b1">s1</font>,<font color="Olive" title="b2">s2</font> st <font color="Olive" title="b4">T</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">h</font> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <font color="Olive" title="b1">s1</font></span>)</span> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">h</font> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <font color="Olive" title="b2">s2</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Olive" title="b3">t</font> )</span><br/>





<span class="kw">assume </span><a NAME="E1:37"/><span class="lab"><font color="Green" title="E27">A1</font></span>: 
<font color="Maroon" title="c4">h</font> <a href="msualg_3.html#R1" title="MSUALG_3:pred.1">is_homomorphism</a> <font color="Maroon" title="c2">A1</font>,<font color="Maroon" title="c3">A2</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">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>  st  <a href="msualg_6.html#K3" title="MSUALG_6:func.3">TranslationRel</a> <font color="Maroon" title="c1">S</font> <a href="rewrite1.html#R1" title="REWRITE1:pred.1">reduces</a> <font color="Olive" title="b1">s1</font>,<font color="Olive" title="b2">s2</font> holds <br/>for <font color="Olive" title="b3">t</font> being   <a href="msualg_6.html#M2" title="MSUALG_6:mode.2">Translation</a> of <font color="Maroon" title="c2">A1</font>,<font color="Olive" title="b1">s1</font>,<font color="Olive" title="b2">s2</font>  ex <font color="Olive" title="b4">T</font> being   <a href="msualg_6.html#M2" title="MSUALG_6:mode.2">Translation</a> of <font color="Maroon" title="c3">A2</font>,<font color="Olive" title="b1">s1</font>,<font color="Olive" title="b2">s2</font> st <font color="Olive" title="b4">T</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">h</font> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <font color="Olive" title="b1">s1</font></span>)</span> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">h</font> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <font color="Olive" title="b2">s2</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Olive" title="b3">t</font></span><br/>

<span class="kw">defpred </span><font color="Maroon">S<sub>1</sub></font>[  <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a>,  <a href="msualg_1.html#NM1" title="MSUALG_1:NM.1">SortSymbol</a> of <font color="Maroon" title="c1">S</font>,  <a href="msualg_1.html#NM1" title="MSUALG_1:NM.1">SortSymbol</a> of <font color="Maroon" title="c1">S</font>] <span class="kw">means </span> ex <font color="Olive" title="b1">T</font> being   <a href="msualg_6.html#M2" title="MSUALG_6:mode.2">Translation</a> of <font color="Maroon" title="c3">A2</font>,$2,$3 st <font color="Olive" title="b1">T</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">h</font> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> $2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">h</font> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> $3</span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> $1;<br/>
<a NAME="E2:37"/><span class="lab"><font color="Green" title="E28">A2</font></span>: 
for <font color="Olive" title="b1">s1</font>, <font color="Olive" title="b2">s2</font>, <font color="Olive" title="b3">s3</font> being  <a href="msualg_1.html#NM1" title="MSUALG_1:NM.1">SortSymbol</a> of <font color="Maroon" title="c1">S</font>  st  <a href="msualg_6.html#K3" title="MSUALG_6:func.3">TranslationRel</a> <font color="Maroon" title="c1">S</font> <a href="rewrite1.html#R1" title="REWRITE1:pred.1">reduces</a> <font color="Olive" title="b1">s1</font>,<font color="Olive" title="b2">s2</font> holds <br/>for <font color="Olive" title="b4">t</font> being   <a href="msualg_6.html#M2" title="MSUALG_6:mode.2">Translation</a> of <font color="Maroon" title="c2">A1</font>,<font color="Olive" title="b1">s1</font>,<font color="Olive" title="b2">s2</font>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b4">t</font>,<font color="Olive" title="b1">s1</font>,<font color="Olive" title="b2">s2</font>] holds <br/>for <font color="Olive" title="b5">f</font> being  <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a>  st <font color="Olive" title="b5">f</font> <a href="msualg_6.html#R1" title="MSUALG_6:pred.1">is_e.translation_of</a> <font color="Maroon" title="c2">A1</font>,<font color="Olive" title="b2">s2</font>,<font color="Olive" title="b3">s3</font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b5">f</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Olive" title="b4">t</font>,<font color="Olive" title="b1">s1</font>,<font color="Olive" title="b3">s3</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c5">s1</font>, <font color="Maroon" title="c6">s2</font>, <font color="Maroon" title="c7">s3</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"> (  <a href="msualg_6.html#K3" title="MSUALG_6:func.3">TranslationRel</a> <font color="Maroon" title="c1">S</font> <a href="rewrite1.html#R1" title="REWRITE1:pred.1">reduces</a> <font color="Maroon" title="c5">s1</font>,<font color="Maroon" title="c6">s2</font> implies for <font color="Olive" title="b1">t</font> being   <a href="msualg_6.html#M2" title="MSUALG_6:mode.2">Translation</a> of <font color="Maroon" title="c2">A1</font>,<font color="Maroon" title="c5">s1</font>,<font color="Maroon" title="c6">s2</font>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">t</font>,<font color="Maroon" title="c5">s1</font>,<font color="Maroon" title="c6">s2</font>] holds <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="Maroon" title="c2">A1</font>,<font color="Maroon" title="c6">s2</font>,<font color="Maroon" title="c7">s3</font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b2">f</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Olive" title="b1">t</font>,<font color="Maroon" title="c5">s1</font>,<font color="Maroon" title="c7">s3</font>] )</span><br/>

<span class="kw">assume </span><a NAME="E1:37_1"/><span class="lab"><font color="Green" title="E29">A3</font></span>: 
 <a href="msualg_6.html#K3" title="MSUALG_6:func.3">TranslationRel</a> <font color="Maroon" title="c1">S</font> <a href="rewrite1.html#R1" title="REWRITE1:pred.1">reduces</a> <font color="Maroon" title="c5">s1</font>,<font color="Maroon" title="c6">s2</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">t</font> being   <a href="msualg_6.html#M2" title="MSUALG_6:mode.2">Translation</a> of <font color="Maroon" title="c2">A1</font>,<font color="Maroon" title="c5">s1</font>,<font color="Maroon" title="c6">s2</font>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">t</font>,<font color="Maroon" title="c5">s1</font>,<font color="Maroon" title="c6">s2</font>] holds <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="Maroon" title="c2">A1</font>,<font color="Maroon" title="c6">s2</font>,<font color="Maroon" title="c7">s3</font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b2">f</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Olive" title="b1">t</font>,<font color="Maroon" title="c5">s1</font>,<font color="Maroon" title="c7">s3</font>]</span><br/>

<span class="kw">let </span><font color="Maroon" title="c8">t</font> be    <a href="msualg_6.html#M2" title="MSUALG_6:mode.2">Translation</a> of <font color="Maroon" title="c2">A1</font>,<font color="Maroon" title="c5">s1</font>,<font color="Maroon" title="c6">s2</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">S<sub>1</sub></font>[<font color="Maroon" title="c8">t</font>,<font color="Maroon" title="c5">s1</font>,<font color="Maroon" title="c6">s2</font>] implies 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="c2">A1</font>,<font color="Maroon" title="c6">s2</font>,<font color="Maroon" title="c7">s3</font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">f</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c8">t</font>,<font color="Maroon" title="c5">s1</font>,<font color="Maroon" title="c7">s3</font>] )</span><br/>

<span class="kw">given </span><font color="Maroon" title="c9">T</font> being    <a href="msualg_6.html#M2" title="MSUALG_6:mode.2">Translation</a> of <font color="Maroon" title="c3">A2</font>,<font color="Maroon" title="c5">s1</font>,<font color="Maroon" title="c6">s2</font><span class="kw"> such that </span><a NAME="E3:37_1"/><span class="lab"><font color="Green" title="E30">A4</font></span>: 
<font color="Maroon" title="c9">T</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">h</font> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <font color="Maroon" title="c5">s1</font></span>)</span> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">h</font> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <font color="Maroon" title="c6">s2</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c8">t</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">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="c2">A1</font>,<font color="Maroon" title="c6">s2</font>,<font color="Maroon" title="c7">s3</font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">f</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c8">t</font>,<font color="Maroon" title="c5">s1</font>,<font color="Maroon" title="c7">s3</font>]</span><br/>

<span class="kw">let </span><font color="Maroon" title="c10">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="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c10">f</font> <a href="msualg_6.html#R1" title="MSUALG_6:pred.1">is_e.translation_of</a> <font color="Maroon" title="c2">A1</font>,<font color="Maroon" title="c6">s2</font>,<font color="Maroon" title="c7">s3</font> implies <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c10">f</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c8">t</font>,<font color="Maroon" title="c5">s1</font>,<font color="Maroon" title="c7">s3</font>] )</span><br/>

<span class="kw">assume </span><a NAME="E4:37_1"/>
<font color="Maroon" title="c10">f</font> <a href="msualg_6.html#R1" title="MSUALG_6:pred.1">is_e.translation_of</a> <font color="Maroon" title="c2">A1</font>,<font color="Maroon" title="c6">s2</font>,<font color="Maroon" title="c7">s3</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">S<sub>1</sub></font>[<font color="Maroon" title="c10">f</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c8">t</font>,<font color="Maroon" title="c5">s1</font>,<font color="Maroon" title="c7">s3</font>]</span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c11">T1</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <span class="p1">(<span class="default">the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Maroon" title="c3">A2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">s2</font></span>)</span>,<span class="p1">(<span class="default">the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Maroon" title="c3">A2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">s3</font></span>)</span><span class="kw"> such that </span><br/><a NAME="E6:37_1"/><span class="lab"><font color="Green" title="E31">A5</font></span>: 
<font color="Maroon" title="c11">T1</font> <a href="msualg_6.html#R1" title="MSUALG_6:pred.1">is_e.translation_of</a> <font color="Maroon" title="c3">A2</font>,<font color="Maroon" title="c6">s2</font>,<font color="Maroon" title="c7">s3</font>
 <span class="kw">and </span><br/><a NAME="E7:37_1"/><span class="lab"><font color="Green" title="E32">A6</font></span>: 
<font color="Maroon" title="c11">T1</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">h</font> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <font color="Maroon" title="c6">s2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">h</font> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <font color="Maroon" title="c7">s3</font></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c10">f</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:37"><span class="lab"><font color="Green" title="E27">A1</font></span></a>, <a class="ref" href="msualg_6.html#T23" target="_self" title="MSUALG_6:th.23">Th23</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c12">T2</font> = <font color="Maroon" title="c11">T1</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c9">T</font> as    <a href="msualg_6.html#M2" title="MSUALG_6:mode.2">Translation</a> of <font color="Maroon" title="c3">A2</font>,<font color="Maroon" title="c5">s1</font>,<font color="Maroon" title="c7">s3</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:37_1"><span class="lab"><font color="Green" title="E29">A3</font></span></a>, <a class="txt" href="#E6:37_1"><span class="lab"><font color="Green" title="E31">A5</font></span></a>, <a class="ref" href="msualg_6.html#T19" target="_self" title="MSUALG_6:th.19">Th19</a></span>;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c12">T2</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="c12">T2</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">h</font> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <font color="Maroon" title="c5">s1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">h</font> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <font color="Maroon" title="c7">s3</font></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">f</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c8">t</font></span>)</span></span><br/>

<span class="kw">thus </span><font color="Maroon" title="c12">T2</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">h</font> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <font color="Maroon" title="c5">s1</font></span>)</span> = 
<font color="Maroon" title="c11">T1</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">T</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">h</font> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <font color="Maroon" title="c5">s1</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#T55" title="RELAT_1:th.55">RELAT_1:55</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">h</font> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <font color="Maroon" title="c7">s3</font></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c10">f</font></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c8">t</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:37_1"><span class="lab"><font color="Green" title="E30">A4</font></span></a>, <a class="txt" href="#E7:37_1"><span class="lab"><font color="Green" title="E32">A6</font></span></a>, <a class="ref" href="relat_1.html#T55" title="RELAT_1:th.55">RELAT_1:55</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">h</font> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <font color="Maroon" title="c7">s3</font></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">f</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c8">t</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#T55" title="RELAT_1:th.55">RELAT_1:55</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/>


</div><span class="kw">end;</span></div>
<a NAME="E3:37"/><span class="lab"><font color="Green" title="E29">A7</font></span>: 
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> holds  <font color="Maroon">S<sub>1</sub></font>[ <a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a> <span class="p1">(<span class="default">the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Maroon" title="c2">A1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">s</font></span>)</span>,<font color="Olive" title="b1">s</font>,<font color="Olive" title="b1">s</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><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>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>1</sub></font>[ <a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a> <span class="p1">(<span class="default">the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Maroon" title="c2">A1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">s</font></span>)</span>,<font color="Maroon" title="c5">s</font>,<font color="Maroon" title="c5">s</font>]</span><br/>

<a NAME="E1:37_2"/><span class="lab"><font color="Green" title="E30">A8</font></span>: 
<span class="p1">(<span class="default"><a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a> <span class="p2">(<span class="default">the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Maroon" title="c3">A2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">s</font></span>)</span></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">h</font> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <font color="Maroon" title="c5">s</font></span>)</span> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c4">h</font> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <font color="Maroon" title="c5">s</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#T23" title="FUNCT_2:th.23">FUNCT_2:23</a></span>;<br/>
<a NAME="E2:37_2"/><span class="lab"><font color="Green" title="E31">A9</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">h</font> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <font color="Maroon" title="c5">s</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p1">(<span class="default"><a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a> <span class="p2">(<span class="default">the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Maroon" title="c2">A1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">s</font></span>)</span></span>)</span> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c4">h</font> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <font color="Maroon" title="c5">s</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#T23" title="FUNCT_2:th.23">FUNCT_2:23</a></span>;<br/>
<a NAME="E3:37_2"/>
 <a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a> <span class="p1">(<span class="default">the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Maroon" title="c3">A2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">s</font></span>)</span> is    <a href="msualg_6.html#M2" title="MSUALG_6:mode.2">Translation</a> of <font color="Maroon" title="c3">A2</font>,<font color="Maroon" title="c5">s</font>,<font color="Maroon" title="c5">s</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="msualg_6.html#T16" target="_self" title="MSUALG_6:th.16">Th16</a></span>;<br/>
<span class="kw">hence </span><a NAME="E4:37_2"/>
<font color="Maroon">S<sub>1</sub></font>[ <a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a> <span class="p1">(<span class="default">the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Maroon" title="c2">A1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">s</font></span>)</span>,<font color="Maroon" title="c5">s</font>,<font color="Maroon" title="c5">s</font>]
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:37_2"><span class="lab"><font color="Green" title="E30">A8</font></span></a>, <a class="txt" href="#E2:37_2"><span class="lab"><font color="Green" title="E31">A9</font></span></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/>


</div><span class="kw">end;</span></div>
<span class="kw">thus </span><a NAME="E4:37"/>
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>  st  <a href="msualg_6.html#K3" title="MSUALG_6:func.3">TranslationRel</a> <font color="Maroon" title="c1">S</font> <a href="rewrite1.html#R1" title="REWRITE1:pred.1">reduces</a> <font color="Olive" title="b1">s1</font>,<font color="Olive" title="b2">s2</font> holds <br/>for <font color="Olive" title="b3">t</font> being   <a href="msualg_6.html#M2" title="MSUALG_6:mode.2">Translation</a> of <font color="Maroon" title="c2">A1</font>,<font color="Olive" title="b1">s1</font>,<font color="Olive" title="b2">s2</font> holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b3">t</font>,<font color="Olive" title="b1">s1</font>,<font color="Olive" title="b2">s2</font>]
 <span class="kw">from</span> <span class="lab"><a class="ref" href="msualg_6.html#S1" target="_self" title="MSUALG_6:sch.1">MSUALG_6:sch 1</a>(<a class="txt" href="#E3:37"><span class="lab"><font color="Green" title="E29">A7</font></span></a>, <a class="txt" href="#E2:37"><span class="lab"><font color="Green" title="E28">A2</font></span></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/>


</div>
