reserve a,b,c,d,e,z for object, A,B,C,D,E for set;

theorem
 for x,y,X,Y,Z being set
  st x`1_3 = y`1_3 & x`2_3 = y`2_3 & x`3_3 = y`3_3 &
  y in [:X,Y,Z:] & x in [:X,Y,Z:]
     holds x = y
  proof
   let x,y,X,Y,Z be set;
   assume
A1:  x`1_3 = y`1_3 & x`2_3 = y`2_3 & x`3_3 = y`3_3 & y in [:X,Y,Z:];
   assume x in [:X,Y,Z:];
   hence x = [ x`1_3, x`2_3, x`3_3 ] by Th3
      .= y by A1,Th3;
  end;
