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:];

theorem
 for X1,X2,X3,X4 being non empty set
 for x being Element of [:X1,X2,X3,X4:]
  st for xx1 being Element of X1,
             xx2 being Element of X2,
             xx3 being Element of X3,
             xx4 being Element of X4 st x = [xx1,
  xx2,xx3,xx4] holds y4 = xx4
 holds y4 =x`4_4
proof
 let X1,X2,X3,X4 be non empty set;
 let x be Element of [:X1,X2,X3,X4:];
  assume that
A1: for xx1 being Element of X1,
             xx2 being Element of X2,
             xx3 being Element of X3,
             xx4 being Element of X4 st x = [xx1,xx2,xx3,xx4] holds y4 = xx4;
  x = [x`1_4,x`2_4,x`3_4,x`4_4];
  hence thesis by A1;
end;
