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,A3,A4:] = { [x1,x2,x3,x4] : x1 in A1 & x2 in A2 & x3 in A3 &
  x4 in A4 }
proof
  thus [:A1,A2,A3,A4:] c= { [x1,x2,x3,x4] : x1 in A1 & x2 in A2 & x3 in A3 &
  x4 in A4 }
  proof
    let a be object;
    assume
A1: a in [:A1,A2,A3,A4:];
    reconsider A1 as non empty Subset of X1 by A1,MCART_1:51;
    reconsider A2 as non empty Subset of X2 by A1,MCART_1:51;
    reconsider A3 as non empty Subset of X3 by A1,MCART_1:51;
    reconsider A4 as non empty Subset of X4 by A1,MCART_1:51;
A2: a in [:A1,A2,A3,A4:] by A1;
    then reconsider x = a as Element of [:X1,X2,X3,X4:];
A3: x`3_4 in A3 & x`4_4 in A4 by A2,MCART_1:83;
A4: x = [x`1_4,x`2_4,x`3_4,x`4_4];
    x`1_4 in A1 & x`2_4 in A2 by A2,MCART_1:83;
    hence thesis by A4,A3;
  end;
  let a be object;
  assume a in { [x1,x2,x3,x4] : x1 in A1 & x2 in A2 & x3 in A3 & x4 in A4 };
  then ex x1,x2,x3,x4 st a = [x1,x2,x3,x4] & x1 in A1 & x2 in A2 & x3 in A3 &
  x4 in A4;
  hence thesis by MCART_1:80;
end;
