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
  for X,Y being finite natural-membered set st Y <> {} & (ex x being set
  st x in X & {x} <N= Y) holds (Sgm0 X).0 <= (Sgm0 Y).0
proof
  let X,Y be finite natural-membered set;
  assume that
A1: Y <> {} and
A2: ex x being set st x in X & {x} <N= Y;
  consider x being set such that
A3: x in X and
A4: {x} <N= Y by A2;
  0 <> card Y by A1;
  then 0 < len (Sgm0 Y) by Th20;
  then
A5: 0 in dom (Sgm0 Y) by AFINSQ_1:86;
  rng (Sgm0 Y)=Y by Def4;
  then
A6: (Sgm0 Y).0 in Y by A5,FUNCT_1:def 3;
  reconsider x0=x as Element of NAT by A3,ORDINAL1:def 12;
  set nx=x0;
  nx in {x0} by TARSKI:def 1;
  then
A7: nx<=(Sgm0 Y).0 by A4,A6;
  {x0} c= X
  by A3,TARSKI:def 1;
  then
A8: (Sgm0 X).0 <= (Sgm0 {x0}).0 by Th21;
  (Sgm0 {x0}).0=nx by Th22;
  hence thesis by A8,A7,XXREAL_0:2;
end;
