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 Th36:
  for B being set holds len SubXFinS (p,B)=
   card (B/\ Segm(len p)) &
  for i st i < len SubXFinS (p,B) holds SubXFinS
  (p,B).i=p.((Sgm0 (B/\ Segm(len p))).i)
proof
  let  B be set;
  B/\ Segm len p c= dom p by XBOOLE_1:17;
  then rng Sgm0(B/\ Segm len p) c= dom p by Def4;
  then dom SubXFinS (p,B) = len Sgm0(B/\ Segm len p) by RELAT_1:27
    .= card(B/\ Segm len p) by Th20;
  hence len SubXFinS (p,B)=card (B/\ Segm len p);
  let i;
  assume i < len SubXFinS (p,B);
  hence thesis by FUNCT_1:12,AFINSQ_1:86;
end;
