reserve

  k,n for Nat,
  x,y,X,Y,Z for set;

theorem Th5:
  X \ Y = X \ Z & Y c= X & Z c= X implies Y = Z
proof
  assume that
A1: X \ Y = X \ Z and
A2: Y c= X and
A3: Z c= X;
A4: Y \ Z c= X by A2;
  X \ Z misses Y by A1,XBOOLE_1:106;
  then Y \ Z = {} by A4,XBOOLE_1:67,81;
  then
A5: Y c= Z by XBOOLE_1:37;
A6: Z \ Y c= X by A3;
  X \ Y misses Z by A1,XBOOLE_1:106;
  then Z \ Y = {} by A6,XBOOLE_1:67,81;
  then Z c= Y by XBOOLE_1:37;
  hence thesis by A5,XBOOLE_0:def 10;
end;
