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,
     A1 being non empty Subset of X1, A2 being non empty Subset of X2,
     A3 being non empty Subset of X3, A4 being non empty Subset of X4
  for x being Element of [:X1,X2,X3,X4:] st x in [:A1,A2,A3,A4:]
  holds x`1_4 in A1 & x`2_4 in A2 & x`3_4 in A3 & x`4_4 in A4
proof
 let X1,X2,X3,X4 be non empty set,
     A1 be non empty Subset of X1, A2 be non empty Subset of X2,
     A3 be non empty Subset of X3, A4 be non empty Subset of X4;
  let x be Element of [:X1,X2,X3,X4:];
  assume
 x in [:A1,A2,A3,A4:];
  then reconsider y = x as Element of [:A1,A2,A3,A4:];
A1: y`2_4 in A2;
A2: y`4_4 in A4;
A3: y`3_4 in A3;
  y`1_4 in A1;
  hence thesis by A1,A3,A2;
end;
