theorem
 for X1,X2,X3 being non empty set
 for x being Element of [:X1,X2,X3:]
  st for xx1 being Element of X1,xx2 being Element of X2,
             xx3 being Element of X3 st x = [xx1,xx2,xx3] holds y3 = xx3
 holds y3 =x`3_3
proof
 let X1,X2,X3 be non empty set;
  let x be Element of [:X1,X2,X3:];
  assume that
A1: for xx1 being Element of X1,xx2 being Element of X2,
             xx3 being Element of X3 st x = [xx1,xx2,xx3] holds y3 = xx3;
  x = [x`1_3,x`2_3,x`3_3];
  hence thesis by A1;
end;
