reserve a,b,c,d for set,
  D,X1,X2,X3,X4 for non empty set,
  x1,y1,z1 for Element of X1,
  x2 for Element of X2,
  x3 for Element of X3,
  x4 for Element of X4,
  A1,B1 for Subset of X1;
reserve x,y for Element of [:X1,X2,X3:];
reserve x for Element of [:X1,X2,X3,X4:];
reserve A2 for Subset of X2,
  A3 for Subset of X3,
  A4 for Subset of X4;

theorem
  [:A1,A2:] = { [x1,x2] : x1 in A1 & x2 in A2 }
proof
  thus [:A1,A2:] c= { [x1,x2] : x1 in A1 & x2 in A2 }
  proof
    let a be object;
    assume
A1: a in [:A1,A2:];
    then reconsider x = a as Element of [:X1,X2:];
A2: x = [x`1,x`2];
    x`1 in A1 & x`2 in A2 by A1,MCART_1:10;
    hence thesis by A2;
  end;
  let a be object;
  assume a in { [x1,x2] : x1 in A1 & x2 in A2 };
  then ex x1,x2 st a = [x1,x2] & x1 in A1 & x2 in A2;
  hence thesis by ZFMISC_1:87;
end;
