reserve x,y,z for set;

theorem Th2:
  for S being non void Signature for X being ManySortedSet of the
carrier of S for s being SortSymbol of S holds [x,s] in the carrier of DTConMSA
  X iff x in X.s
proof
  let S be non void Signature;
  let X be ManySortedSet of the carrier of S;
  let s be SortSymbol of S;
A1: DTConMSA X = DTConstrStr (# [:the carrier' of S,{the carrier of S}:] \/
    Union coprod X, REL(X) #) by MSAFREE:def 8;
A2: dom coprod X = the carrier of S by PARTFUN1:def 2;
  s in the carrier of S;
  then s <> the carrier of S;
  then not s in {the carrier of S} by TARSKI:def 1;
  then
A3: not [x,s] in [:the carrier' of S,{the carrier of S}:] by ZFMISC_1:87;
  hereby
    assume [x,s] in the carrier of DTConMSA X;
    then [x,s] in Union coprod X by A1,A3,XBOOLE_0:def 3;
    then consider y being object such that
A4: y in dom coprod X and
A5: [x,s] in (coprod X).y by CARD_5:2;
    (coprod X).y = coprod(y,X) by A4,MSAFREE:def 3;
    then consider z such that
A6: z in X.y and
A7: [x,s] = [z,y] by A4,A5,MSAFREE:def 2;
    x = z by A7,XTUPLE_0:1;
    hence x in X.s by A6,A7,XTUPLE_0:1;
  end;
  assume x in X.s;
  then [x,s] in coprod(s,X) by MSAFREE:def 2;
  then [x,s] in (coprod X).s by MSAFREE:def 3;
  then [x,s] in Union coprod X by A2,CARD_5:2;
  hence thesis by A1,XBOOLE_0:def 3;
end;
