reserve v,x,x1,x2,x3,x4,y,y1,y2,y3,y4,z,z1,z2 for object,
  X,X1,X2,X3,X4,Y,Y1,Y2,Y3,Y4,Y5,
  Z,Z1,Z2,Z3,Z4,Z5 for set;
reserve p for pair object;
reserve R for Relation;
reserve xx1 for Element of X1,
  xx2 for Element of X2,
  xx3 for Element of X3;
reserve xx4 for Element of X4;
reserve A1 for Subset of X1,
  A2 for Subset of X2,
  A3 for Subset of X3,
  A4 for Subset of X4;
reserve x for Element of [:X1,X2,X3:];
reserve x for Element of [:X1,X2,X3,X4:];
reserve x for object;

theorem
 [[x1,x2],[x3,x4]] = [[y1,y2],[y3,y4]] implies x1=y1 & x2=y2 & x3=y3 & x4=y4
proof
 assume [[x1,x2],[x3,x4]] = [[y1,y2],[y3,y4]];
   then [x1,x2] = [y1,y2] & [x3,x4] = [y3,y4] by XTUPLE_0:1;
  hence x1=y1 & x2=y2 & x3=y3 & x4=y4 by XTUPLE_0:1;
end;
