reserve x,x1,x2,x3,x4,y,y1,y2,y3,y4,z,z1,z2,z2,z4 for object;
reserve X,X1,X2,X3,X4,Y for set;

theorem
 for a,b being quadruple object
  st a`1_4 = b`1_4 & a`2_4 = b`2_4 & a`3_4 = b`3_4 & a`4_4 = b`4_4
  holds a = b
proof let a,b be quadruple object such that
A1: a`1_4 = b`1_4 & a`2_4 = b`2_4 & a`3_4 = b`3_4 & a`4_4 = b`4_4;
  a = [a`1_4,a`2_4,a`3_4,a`4_4] & b = [b`1_4,b`2_4,b`3_4,b`4_4];
 hence a = b by A1;
end;
