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

theorem     :: SETWISEO:6
  x in A or x = B implies x in A (\/) {B}
proof
  assume
A1: x in A or x = B;
  let i such that
A2: i in I;
  per cases by A1;
  suppose x in A;
    then x.i in A.i by A2;
    then x.i in A.i \/ {B.i} by ZFMISC_1:136;
    then x.i in A.i \/ {B}.i by A2,Def1;
    hence thesis by A2,PBOOLE:def 4;
  end;
  suppose x = B;
    then x.i in A.i \/ {B.i} by ZFMISC_1:136;
    then x.i in A.i \/ {B}.i by A2,Def1;
    hence thesis by A2,PBOOLE:def 4;
  end;
end;
