theorem Th43:
  for t being Element of A0
  for s being SortSymbol of S st t in (the Sorts of Free(S,X)).s holds
  t in (the Sorts of A0).s
  proof
    let t be Element of A0;
    consider x being object such that
A1: x in dom the Sorts of A0 & t in (the Sorts of A0).x by CARD_5:2;
    reconsider x as SortSymbol of S by A1;
    the Sorts of A0 is ManySortedSubset of the Sorts of Free(S,X)
    by Def6; then
A2: (the Sorts of A0).x c= (the Sorts of Free(S,X)).x by PBOOLE:def 2,def 18;
    let s be SortSymbol of S;
    assume
    t in (the Sorts of Free(S,X)).s;
    hence t in (the Sorts of A0).s by A1,A2,XBOOLE_0:3,PROB_2:def 2;
  end;
