theorem
  [:A1,A2,A3:] = { [x1,x2,x3] : x1 in A1 & x2 in A2 & x3 in A3 }
proof
  thus [:A1,A2,A3:] c= { [x1,x2,x3] : x1 in A1 & x2 in A2 & x3 in A3 }
  proof
    let a be object;
    assume
A1: a in [:A1,A2,A3:];
    reconsider A1 as non empty Subset of X1 by A1,MCART_1:31;
    reconsider A2 as non empty Subset of X2 by A1,MCART_1:31;
    reconsider A3 as non empty Subset of X3 by A1,MCART_1:31;
A2: a in [:A1,A2,A3:] by A1;
    reconsider x = a as Element of [:X1,X2,X3:] by A2;
A3: x = [x`1_3,x`2_3,x`3_3];
A4:  x`3_3 in A3 by A2,MCART_1:72;
    x`1_3 in A1 & x`2_3 in A2 by A2,MCART_1:72;
    hence thesis by A4,A3;
  end;
  let a be object;
  assume a in { [x1,x2,x3] : x1 in A1 & x2 in A2 & x3 in A3 };
  then ex x1,x2,x3 st a= [x1,x2,x3] & x1 in A1 & x2 in A2 & x3 in A3;
  hence thesis by MCART_1:69;
end;
