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:];

theorem
 for X1,X2,X3 being non empty set
 for A1 being non empty Subset of X1, A2 being non empty Subset of X2,
     A3 being non empty Subset of X3
  for x being Element of [:X1,X2,X3:] st x in [:A1,A2,A3:]
   holds x`1_3 in A1 & x`2_3 in A2 & x`3_3 in A3
proof
 let X1,X2,X3 be non empty set;
 let A1 be non empty Subset of X1, A2 be non empty Subset of X2,
     A3 be non empty Subset of X3;
  let x be Element of [:X1,X2,X3:];
  assume
 x in [:A1,A2,A3:];
  then reconsider y = x as Element of [:A1,A2,A3:];
A1: y`2_3 in A2;
A2: y`3_3 in A3;
  y`1_3 in A1;
  hence thesis by A1,A2;
end;
