reserve x,y for object,X,Y,A,B,C,M for set;
reserve P,Q,R,R1,R2 for Relation;

theorem Th15:
  for A being set, X being Subset of A holds
    X = union {{x} where x is Element of A: x in X}
proof
  let A be set, X be Subset of A;
  thus X c= union {{x} where x is Element of A: x in X}
  proof
    let a be object;
    assume
A1: a in X;
    set Y = {a};
A2: a in Y by TARSKI:def 1;
    Y in {{x} where x is Element of A: x in X} by A1;
    hence thesis by A2,TARSKI:def 4;
  end;
  let a be object;
  assume a in union {{x} where x is Element of A: x in X};
  then consider Y being set such that
A3: a in Y and
A4: Y in {{x} where x is Element of A: x in X} by TARSKI:def 4;
  ex x being Element of A st ( Y = {x})&( x in X) by A4;
  hence thesis by A3,TARSKI:def 1;
end;
