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;
reserve D for non empty set,
  F,G for XFinSequence of D,
  b for BinOp of D,
  d,d1,d2 for Element of D;
reserve F for XFinSequence,
        rF,rF1,rF2 for real-valued XFinSequence,
        r for Real,
        cF,cF1,cF2 for complex-valued XFinSequence,
        c,c1,c2 for Complex;

theorem Th56: :: NUMERAL1:4
  len rF1 = len rF2 &
  (for i st i in dom rF1 holds rF1.i<=rF2.i) implies
  Sum rF1 <= Sum rF2
proof
  set d=rF1,e=rF2;
assume that
A1: len d = len e and
A2: for i st i in dom d holds d.i<=e.i;
reconsider d,e as  XFinSequence of REAL;
A3:  Sum d = addreal "**" d & Sum e = addreal "**" e by Th47;
per cases by NAT_1:14;
  suppose A4:len d >=1;
  consider f being sequence of REAL such that
A5: f.0 = d.0 and
A6: for n st n+1 < len d holds f.(n + 1) = addreal.
  (f.n,d.(n + 1)) and
A7: Sum d = f.(len d-1) by A4,Def8,A3;
  consider g being sequence of REAL such that
A8: g.0 = e.0 and
A9: for n st n+1 < len e holds g.(n + 1) = addreal.
  (g.n,e.(n + 1)) and
A10: Sum e = g.(len e-1) by A4,A1,Def8,A3;
  defpred P[Nat] means $1 in dom d implies f.$1 <= g.$1;
A11: now
    let i;
    assume
A12: P[i];
    thus P[i+1]
    proof
      assume
A13:  i+1 in dom d;
      then
A14:  i+1 < len d by AFINSQ_1:86;
      then
A15:  i < len d by NAT_1:13;
A16:  d.(i+1) <= e.(i+1) by A2,A13;
A17:  f.(i+1) = addreal.(f.i,d.(i + 1)) by A6,A14
        .= f.i + d.(i+1) by BINOP_2:def 9;
      g.(i+1) = addreal.(g.i,e.(i + 1)) by A1,A9,A14
        .= g.i + e.(i+1) by BINOP_2:def 9;
      hence thesis by A12,A15,A17,A16,AFINSQ_1:86,XREAL_1:7;
    end;
  end;
  reconsider ld=len d-1 as Element of NAT by A4,NAT_1:21;
  len d-1 < len d - 0 by XREAL_1:10;
  then
A18: ld in len d by AFINSQ_1:86;
A19: P[0] by A2,A5,A8;
  for i holds P[i] from NAT_1:sch 2(A19,A11);
  hence thesis by A1,A7,A10,A18;
end;
suppose len d=0;
  then Sum d = the_unity_wrt addreal & Sum e = the_unity_wrt addreal
     by Def8,A3,A1;
  hence thesis;
end;
end;
