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 Th33: ::from FINSEQ_3:46
  for l,m,n,k being Nat,X being finite natural-membered set st k <
  l & m < len(Sgm0 X) & (Sgm0(X)).m = k & (Sgm0(X)).n = l holds m < n
proof
  let l,m,n,k be Nat,X being finite natural-membered set;
  assume that
A1: k < l and
A2: m < len(Sgm0 X) and
A3: (Sgm0(X)).m = k and
A4: (Sgm0(X)).n = l and
A5: not m < n;
  n < m by A1,A3,A4,A5,XXREAL_0:1;
  hence thesis by A1,A2,A3,A4,Def4;
end;
