reserve u,v,x,x1,x2,y,y1,y2,z,p,a for object,
        A,B,X,X1,X2,X3,X4,Y,Y1,Y2,Z,N,M for set;

theorem Th31:
  {x1,x2} c= Z iff x1 in Z & x2 in Z
proof
  x1 in {x1,x2} & x2 in {x1,x2} by TARSKI:def 2;
  hence {x1,x2} c= Z implies x1 in Z & x2 in Z;
  assume
A1: x1 in Z & x2 in Z;
  let z;
  assume z in {x1,x2};
  hence thesis by A1,TARSKI:def 2;
end;
