reserve s for set,
  i,j for natural Number,
  k for Nat,
  x,x1,x2,x3 for Real,
  r,r1,r2,r3,r4 for Real,
  F,F1,F2,F3 for real-valued FinSequence,
  R,R1,R2 for Element of i-tuples_on REAL;

theorem Th82:
  for R1,R2 being i-element real-valued FinSequence holds
  (for j be Nat st j in Seg i holds R1.j <= R2.j) implies Sum R1 <= Sum R2
proof
  let R1,R2 be i-element real-valued FinSequence;
A0: i is Nat by TARSKI:1;
  defpred P[Nat] means for R1,R2 being $1-element real-valued FinSequence st
  for j be Nat st j in Seg $1 holds R1.j <= R2.j holds Sum R1 <= Sum R2;
A1: for i be Nat st P[i] holds P[i+1]
  proof
    let i be Nat such that
A2: for R1,R2 being i-element real-valued FinSequence st
    for j be Nat st j in Seg i holds R1.j <= R2.j holds Sum R1 <= Sum R2;
    set n = i+1;
    let R1,R2 be n-element real-valued FinSequence such that
A3: for j be Nat st j in Seg n holds R1.j <= R2.j;
    R1 is Element of n-tuples_on REAL by Lm;
    then consider R19 being (Element of i-tuples_on REAL),
     x1 being Element of REAL such that
A4: R1 = R19^<*x1*> by FINSEQ_2:117;
    R2 is Element of n-tuples_on REAL by Lm;
    then consider R29 being (Element of i-tuples_on REAL),
     x2 being Element of REAL such that
A5: R2 = R29^<*x2*> by FINSEQ_2:117;
    for j be Nat st j in Seg i holds R19.j <= R29.j
    proof
      let j be Nat such that
A6:   j in Seg i;
      Seg len R29 = dom R29 & len R29 = i by CARD_1:def 7,FINSEQ_1:def 3;
      then
A7:   R29.j = R2.j by A5,A6,FINSEQ_1:def 7;
      Seg len R19 = dom R19 & len R19 = i by CARD_1:def 7,FINSEQ_1:def 3;
      then R19.j = R1.j by A4,A6,FINSEQ_1:def 7;
      hence thesis by A3,A6,A7,FINSEQ_2:8;
    end;
    then
A8: Sum R19 <= Sum R29 by A2;
A9: R2.n = x2 by A5,FINSEQ_2:116;
    R1.n = x1 by A4,FINSEQ_2:116;
    then
A10: x1 <= x2 by A3,A9,FINSEQ_1:4;
A11: Sum R2 = Sum R29 + x2 by A5,Th74;
    Sum R1 = Sum R19 + x1 by A4,Th74;
    hence thesis by A10,A8,A11,XREAL_1:7;
  end;
A12: P[0];
  for i be Nat holds P[i] from NAT_1:sch 2(A12,A1);
  hence thesis by A0;
end;
