reserve i,j,k,n,m for Nat,
  x,y,z,y1,y2 for object, X,Y,D for set,
  p,q for XFinSequence;
reserve k1,k2 for Nat;

theorem Th21:
  for X,Y being finite natural-membered set st X c= Y & X <> {}
  holds (Sgm0 Y).0 <= (Sgm0 X).0
proof
  let X,Y be finite natural-membered set;
  assume that
A1: X c= Y and
A2: X <> {};
  reconsider X0=X as finite set;
  0 <> card X0 by A2;
  then 0 < len (Sgm0 X) by Th20;
  then
A3: 0 in dom (Sgm0 X) by AFINSQ_1:86;
A4: rng (Sgm0 Y)=Y by Def4;
  rng (Sgm0 X)=X by Def4;
  then (Sgm0 X).0 in X by A3,FUNCT_1:def 3;
  then consider x being object such that
A5: x in dom (Sgm0 Y) and
A6: (Sgm0 Y).x=(Sgm0 X).0 by A1,A4,FUNCT_1:def 3;
  reconsider nx=x as Nat by A5;
A7: nx <len (Sgm0 Y) by A5,AFINSQ_1:86;
  now
    per cases;
    case
      0<>nx;
      hence thesis by A6,A7,Def4;
    end;
    case
      0=nx;
      hence thesis by A6;
    end;
  end;
  hence thesis;
end;
