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;
