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:8
  A (\/) {x} c= B iff x in B & A c= B
proof
  thus A (\/) {x} c= B implies x in B & A c= B
  proof
    assume
A1: A (\/) {x} c= B;
A2: now
      let i;
      assume
A3:   i in I;
      then (A (\/) {x}).i c= B.i by A1;
      then A.i \/ {x}.i c= B.i by A3,PBOOLE:def 4;
      hence A.i \/ {x.i} c= B.i by A3,Def1;
    end;
    thus x in B
    proof
      let i;
      assume i in I;
      then A.i \/ {x.i} c= B.i by A2;
      hence thesis by ZFMISC_1:137;
    end;
    let i;
    assume i in I;
    then A.i \/ {x.i} c= B.i by A2;
    hence thesis by ZFMISC_1:137;
  end;
  assume that
A4: x in B and
A5: A c= B;
  let i;
  assume
A6: i in I;
  then
A7: x.i in B.i by A4;
  A.i c= B.i by A5,A6;
  then A.i \/ {x.i} c= B.i by A7,ZFMISC_1:137;
  then A.i \/ {x}.i c= B.i by A6,Def1;
  hence thesis by A6,PBOOLE:def 4;
end;
