reserve x,A,B,X,X9,Y,Y9,Z,V for set;

theorem
  X \/ Y = Z \/ Y & X misses Y & Z misses Y implies X = Z
proof
  assume that
A1: X \/ Y = Z \/ Y and
A2: X /\ Y = {} and
A3: Z /\ Y = {};
  thus X c= Z
  proof
    let x be object such that
A4: x in X;
    X c= Z \/ Y by A1,Th7;
    then
A5: x in Z \/ Y by A4;
    not x in Y by A2,A4,XBOOLE_0:def 4;
    hence thesis by A5,XBOOLE_0:def 3;
  end;
  let x be object such that
A6: x in Z;
  Z c= X \/ Y by A1,Th7;
  then
A7: x in X \/ Y by A6;
  not x in Y by A3,A6,XBOOLE_0:def 4;
  hence thesis by A7,XBOOLE_0:def 3;
end;
