reserve x,y,z for set;

theorem Th22:
  for S being non void Signature for Y being non-empty
ManySortedSet of the carrier of S for X being ManySortedSet of the carrier of S
  holds (Reverse Y)""X c= S-Terms(X,Y)
proof
  let S be non void Signature;
  let Y be non-empty ManySortedSet of the carrier of S;
  let X be ManySortedSet of the carrier of S;
  let s be object;
  assume s in the carrier of S;
  then reconsider s9 = s as SortSymbol of S;
  let x be object;
  assume x in ((Reverse Y)""X).s;
  then
A1: x in ((Reverse Y).s9)"(X.s9) by EQUATION:def 1;
  then
A2: x in dom ((Reverse Y).s) by FUNCT_1:def 7;
A3: ((Reverse Y).s).x in X.s by A1,FUNCT_1:def 7;
A4: (Reverse Y).s = Reverse(s9, Y) by MSAFREE:def 18;
  then
A5: dom ((Reverse Y).s) = FreeGen(s9, Y) by FUNCT_2:def 1;
  then consider b being set such that
A6: b in Y.s9 and
A7: x = root-tree [b,s9] by A2,MSAFREE:def 15;
  FreeGen(s9, Y) = {root-tree t where t is Symbol of DTConMSA(Y): t in
  Terminals DTConMSA(Y) & t`2 = s} by MSAFREE:13;
  then consider a being Symbol of DTConMSA(Y) such that
A8: x = root-tree a and
  a in Terminals DTConMSA(Y) and
  a`2 = s by A2,A5;
  Reverse(s9, Y).x = a`1 by A2,A5,A8,MSAFREE:def 17
    .= [b,s]`1 by A8,A7,TREES_4:4
    .= b;
  hence thesis by A3,A4,A6,A7,Th18;
end;
