reserve E, x, y, X for set;
reserve A, B, C, D for Subset of E^omega;
reserve a, a1, a2, b, c, c1, c2, d, ab, bc for Element of E^omega;
reserve e for Element of E;
reserve i, j, k, l, n, n1, n2, m for Nat;

theorem Th18:
  (A ^^ B) ^^ C = A ^^ (B ^^ C)
proof
  now
    let x be object;
    thus x in (A ^^ B) ^^ C implies x in A ^^ (B ^^ C)
    proof
      assume x in (A ^^ B) ^^ C;
      then consider ab, c such that
A1:   ab in (A ^^ B) and
A2:   c in C & x = ab ^ c by Def1;
      consider a, b such that
A3:   a in A and
A4:   b in B & ab = a ^ b by A1,Def1;
      x = a ^ (b ^ c) & b ^ c in B ^^ C by A2,A4,Def1,AFINSQ_1:27;
      hence thesis by A3,Def1;
    end;
    assume x in A ^^ (B ^^ C);
    then consider a, bc such that
A5: a in A and
A6: bc in (B ^^ C) and
A7: x = a ^ bc by Def1;
    consider b, c such that
A8: b in B and
A9: c in C and
A10: bc = b ^ c by A6,Def1;
    x = (a ^ b) ^ c & a ^ b in A ^^ B by A5,A7,A8,A10,Def1,AFINSQ_1:27;
    hence x in (A ^^ B) ^^ C by A9,Def1;
  end;
  hence thesis by TARSKI:2;
end;
