let C be initialized standardized ConstructorSignature; :: thesis: for e being expression of C st (e . {} ) `1 in Constructors holds
e in the Sorts of (Free C,(MSVars C)) . (((e . {} ) `1 ) `1 )
let e be expression of C; :: thesis: ( (e . {} ) `1 in Constructors implies e in the Sorts of (Free C,(MSVars C)) . (((e . {} ) `1 ) `1 ) )
assume A1: (e . {} ) `1 in Constructors ; :: thesis: e in the Sorts of (Free C,(MSVars C)) . (((e . {} ) `1 ) `1 )

per cases ( ex x being Element of Vars st
( e = x -term C & e . {} = [x,a_Term ] ) or ex o being OperSymbol of C st
( e . {} = [o,the carrier of C] & ( o in Constructors or o = * or o = non_op ) ) ) by Th49;

suppose ex x being Element of Vars st
( e = x -term C & e . {} = [x,a_Term ] ) ; :: thesis: e in the Sorts of (Free C,(MSVars C)) . (((e . {} ) `1 ) `1 )

then consider x being Element of Vars such that
A2: ( e = x -term C & e . {} = [x,a_Term ] ) ;
(e . {} ) `1 = x by A2, MCART_1:7;
hence e in the Sorts of (Free C,(MSVars C)) . (((e . {} ) `1 ) `1 ) by A1, Lem2, XBOOLE_0:3; :: thesis: verum

end;

suppose ex o being OperSymbol of C st
( e . {} = [o,the carrier of C] & ( o in Constructors or o = * or o = non_op ) ) ; :: thesis: e in the Sorts of (Free C,(MSVars C)) . (((e . {} ) `1 ) `1 )

then consider o being OperSymbol of C such that
A3: e . {} = [o,the carrier of C] ;
A4: (e . {} ) `1 = o by A3, MCART_1:7;
( * in {* ,non_op } & non_op in {* ,non_op } ) by TARSKI:def 2;
then ( o <> * & o <> non_op ) by A1, A4, XBOOLE_0:3, ABCMIZ_1:39;
then A5: o is constructor by ABCMIZ_1:def 11;
set X = MSVars C;
reconsider t = e as Term of C,((MSVars C) \/ (the carrier of C --> {0 })) by MSAFREE3:9;
A6: the_sort_of t = the_result_sort_of o by A3, MSATERM:17
.= o `1 by A5, StandardizedDef ;
variables_in t c= MSVars C by MSAFREE3:28;
then e in { t1 where t1 is Term of C,((MSVars C) \/ (the carrier of C --> {0 })) : ( the_sort_of t1 = the_sort_of t & variables_in t1 c= MSVars C ) } ;
then e in (C -Terms (MSVars C),((MSVars C) \/ (the carrier of C --> {0 }))) . (the_sort_of t) by MSAFREE3:def 6;
hence e in the Sorts of (Free C,(MSVars C)) . (((e . {} ) `1 ) `1 ) by A4, A6, MSAFREE3:24; :: thesis: verum

end;

end;