reserve S for OrderSortedSign;
reserve S for OrderSortedSign,
  X for ManySortedSet of S,
  o for OperSymbol of S ,
  b for Element of ([:the carrier' of S,{the carrier of S}:] \/ Union (coprod X
  ))*;
reserve x for set;

theorem
  for S being OrderSortedSign, X being non-empty ManySortedSet of S, s
being Element of S, x being set st x in X.s for t being Element of TS DTConOSA(
  X) st t = root-tree [x,s] holds LeastSort t = s
proof
  let S being OrderSortedSign, X being non-empty ManySortedSet of S, s being
  Element of S, x being set such that
A1: x in X.s;
  reconsider s2 = s as Element of S;
  let t being Element of TS DTConOSA(X) such that
A2: t = root-tree [x,s];
A3: for s1 being Element of S st t in (the Sorts of ParsedTermsOSA(X)).s1
  holds s2 <= s1 by A1,A2,Th10;
  t in (the Sorts of ParsedTermsOSA(X)).s2 by A1,A2,Th10;
  hence thesis by A3,Def12;
end;
