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;
reserve e for object, X,X1,X2,Y1,Y2 for set;

theorem
  for A, B being set st A c= B & B c= A \/ {a} holds A \/ {a} = B or A = B
proof
  let A, B be set;
  assume that
A1: A c= B and
A2: B c= A \/ {a};
  assume that
A3: A \/ {a} <> B and
A4: A <> B;
 not a in B
  proof
    assume a in B;
    then {a} c= B by Lm1;
    hence thesis by A1,A2,A3,XBOOLE_1:8;
  end;
  hence thesis by A2,Th134,A1,A4;
end;
