reserve a,b,c,d for set,
  D,X1,X2,X3,X4 for non empty set,
  x1,y1,z1 for Element of X1,
  x2 for Element of X2,
  x3 for Element of X3,
  x4 for Element of X4,
  A1,B1 for Subset of X1;
reserve x,y for Element of [:X1,X2,X3:];
reserve x for Element of [:X1,X2,X3,X4:];

theorem
  for x,y being Element of [:X1,X2,X3,X4:]
   st x`1_4=y`1_4 & x`2_4=y`2_4 & x`3_4=y
  `3_4 & x`4_4=y`4_4 holds x=y
proof
  let x,y be Element of [:X1,X2,X3,X4:];
  [x`1_4,x`2_4,x`3_4,x`4_4]=x;
  hence thesis by AB;
end;
