reserve x,y,y1,y2 for set;
reserve G for Group;
reserve a,b,c,d,g,h for Element of G;
reserve A,B,C,D for Subset of G;
reserve H,H1,H2,H3 for Subgroup of G;
reserve n for Nat;
reserve i for Integer;

theorem
  A" |^ B = (A |^ B)"
proof
  thus A" |^ B c= (A |^ B)"
  proof
    let x be object;
    assume x in A" |^ B;
    then consider a,b such that
A1: x = a |^ b and
A2: a in A" and
A3: b in B;
    consider c such that
A4: a = c" & c in A by A2;
    x = (c |^ b)" & c |^ b in A |^ B by A1,A3,A4,Th26;
    hence thesis;
  end;
  let x be object;
  assume x in (A |^ B)";
  then consider a such that
A5: x = a" and
A6: a in A |^ B;
  consider b,c such that
A7: a = b |^ c and
A8: b in A and
A9: c in B by A6;
A10: b" in A" by A8;
  x = b" |^ c by A5,A7,Th26;
  hence thesis by A9,A10;
end;
