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

theorem     :: ZFMISC_1:38
  {x1,x2} c= X iff x1 in X & x2 in X
proof
  thus {x1,x2} c= X implies x1 in X & x2 in X
  proof
    assume
A1: {x1,x2} c= X;
    thus x1 in X
    proof
      let i;
      assume
A2:   i in I;
      then {x1,x2}.i c= X.i by A1;
      then {x1.i,x2.i} c= X.i by A2,Def2;
      hence thesis by ZFMISC_1:32;
    end;
    let i;
    assume
A3: i in I;
    then {x1,x2}.i c= X.i by A1;
    then {x1.i,x2.i} c= X.i by A3,Def2;
    hence thesis by ZFMISC_1:32;
  end;
  assume that
A4: x1 in X and
A5: x2 in X;
  let i;
  assume
A6: i in I;
  then
A7: x1.i in X.i by A4;
  x2.i in X.i by A5,A6;
  then {x1.i,x2.i} c= X.i by A7,ZFMISC_1:32;
  hence thesis by A6,Def2;
end;
