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 Th4:
  for S be OrderSortedSign, X be non-empty ManySortedSet of S, t be
set holds t in Terminals DTConOSA(X) iff ex s be Element of S, x be set st x in
  X.s & t = [x,s]
proof
  let S be OrderSortedSign, X be non-empty ManySortedSet of S, t be set;
  set D = DTConOSA(X);
A1: Terminals D = Union (coprod X) by Th3
    .= union rng (coprod X) by CARD_3:def 4;
  thus t in Terminals D implies ex s be Element of S, x be set st x in X.s & t
  = [x,s]
  proof
    assume t in Terminals D;
    then consider A be set such that
A2: t in A and
A3: A in rng(coprod X) by A1,TARSKI:def 4;
    consider s be object such that
A4: s in dom (coprod X) and
A5: (coprod X).s = A by A3,FUNCT_1:def 3;
    reconsider s as Element of S by A4;
    (coprod X).s = coprod(s,X) by MSAFREE:def 3;
    then consider x be set such that
A6: x in X.s and
A7: t = [x,s] by A2,A5,MSAFREE:def 2;
    take s;
    take x;
    thus thesis by A6,A7;
  end;
  given s be Element of S, x be set such that
A8: x in X.s and
A9: t = [x,s];
  t in coprod(s,X) by A8,A9,MSAFREE:def 2;
  then
A10: t in (coprod X).s by MSAFREE:def 3;
  dom(coprod X) = the carrier of S by PARTFUN1:def 2;
  then (coprod X).s in rng (coprod X) by FUNCT_1:def 3;
  hence thesis by A1,A10,TARSKI:def 4;
end;
