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 Th28:
  [:{x},{y}:] = {[x,y]}
proof
  now
    let z;
    thus z in [:{x},{y}:] implies z in {[x,y]}
    proof
      assume z in [:{x},{y}:];
      then consider x1,y1 such that
A1:   x1 in {x} & y1 in {y} and
A2:   z=[x1,y1] by Def2;
      x1=x & y1=y by A1,TARSKI:def 1;
      hence thesis by A2,TARSKI:def 1;
    end;
    assume z in {[x,y]};
    then
A3: z=[x,y] by TARSKI:def 1;
    x in {x} & y in {y} by TARSKI:def 1;
    hence z in [:{x},{y}:] by A3,Lm16;
  end;
  hence thesis by TARSKI:2;
end;
