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 Th32:
  for X,Y being finite natural-membered set st Y <> {} & X <N< Y
  holds (Sgm0 Y).0 = (Sgm0 (X \/ Y)).(len (Sgm0 X))
proof
  let X,Y be finite natural-membered set;
  assume that
A1: Y <> {} and
A2: X <N< Y;
  card Y <> 0 by A1;
  then 0<len (Sgm0 Y) by Th20;
  then
  (Sgm0 Y).0 = (Sgm0 (X \/ Y)).((0 qua Element of NAT)+len (Sgm0 X)) by A2,Th31
;
  hence thesis;
end;
