reserve i for object, I for set,
  f for Function,
  x, x1, x2, y, A, B, X, Y, Z for ManySortedSet of I;

theorem     :: BORSUK_1:2
  A in [|x,y|] & A in [|X,Y|] implies A in [|x (/\) X, y (/\) Y|]
proof
  assume that
A1: A in [|x,y|] and
A2: A in [|X,Y|];
  let i;
  assume
A3: i in I;
  then
A4: A.i in [|x,y|].i by A1;
A5: A.i in [|X,Y|].i by A2,A3;
A6: A.i in [:x.i,y.i:] by A3,A4,PBOOLE:def 16;
  A.i in [:X.i,Y.i:] by A3,A5,PBOOLE:def 16;
  then A.i in [:x.i /\ X.i, y.i /\ Y.i:] by A6,ZFMISC_1:113;
  then A.i in [:(x (/\) X).i, y.i /\ Y.i:] by A3,PBOOLE:def 5;
  then A.i in [:(x (/\) X).i, (y (/\) Y).i:] by A3,PBOOLE:def 5;
  hence thesis by A3,PBOOLE:def 16;
end;
