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
  [x,y] in [:{z},Y:] iff x = z & y in Y
proof
A1: x in {z} iff x=z by TARSKI:def 1;
  hence [x,y] in [:{z},Y:] implies x=z & y in Y by Lm16;
  thus thesis by A1,Lm16;
end;
