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;
