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 Th83:
  for R1,R2 being i-element real-valued FinSequence holds
  (for j be Nat st j in Seg i holds R1.j <= R2.j) &
  (ex j be Nat st j in Seg i & 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 be $1-element real-valued FinSequence st
  (for j be Nat st j in Seg $1 holds R1.j <= R2.j) &
  (ex j be Nat st j in Seg $1 & R1.j < R2.j) holds Sum
  R1 < Sum R2;
  now
    let i be Nat such that
A1: P[i];
    let R1,R2 be (i+1)-element real-valued FinSequence such that
A2: for j be Nat st j in Seg (i+1) holds R1.j <= R2.j;
    given j be Nat such that
A3: j in Seg (i+1) and
A4: R1.j < R2.j;
    R1 is Element of (i+1)-tuples_on REAL by Lm;
    then consider R19 being (Element of i-tuples_on REAL),
      x1 being Element of REAL such that
A5: R1 = R19^<*x1*> by FINSEQ_2:117;
    R2 is Element of (i+1)-tuples_on REAL by Lm;
    then consider R29 being (Element of i-tuples_on REAL),
      x2 being Element of REAL such that
A6: R2 = R29^<*x2*> by FINSEQ_2:117;
A7: for j be Nat st j in Seg i holds R19.j <= R29.j
    proof
      let j be Nat such that
A8:   j in Seg i;
      Seg len R29 = dom R29 & len R29 = i by CARD_1:def 7,FINSEQ_1:def 3;
      then
A9:   R29.j = R2.j by A6,A8,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 A5,A8,FINSEQ_1:def 7;
      hence thesis by A2,A8,A9,FINSEQ_2:8;
    end;
A10: R2.(i+1) = x2 by A6,FINSEQ_2:116;
A11: R1.(i+1) = x1 by A5,FINSEQ_2:116;
    now
      per cases by A3,FINSEQ_2:7;
      suppose
A12:    j in Seg i;
        Seg len R29 = dom R29 & len R29 = i by CARD_1:def 7,FINSEQ_1:def 3;
        then
A13:    R29.j = R2.j by A6,A12,FINSEQ_1:def 7;
A14:    Sum R1 = Sum R19 + x1 & Sum R2 = Sum R29 + x2 by A5,A6,Th74;
A15:    x1 <= x2 by A2,A11,A10,FINSEQ_1:4;
        Seg len R19 = dom R19 & len R19 = i by CARD_1:def 7,FINSEQ_1:def 3;
        then R19.j = R1.j by A5,A12,FINSEQ_1:def 7;
        then Sum R19 < Sum R29 by A1,A4,A7,A12,A13;
        hence Sum R1 < Sum R2 by A14,A15,XREAL_1:8;
      end;
      suppose
A16:    j = i+1;
A17:    Sum R2 = Sum R29 + x2 by A6,Th74;
        Sum R19 <= Sum R29 & Sum R1 = Sum R19 + x1 by A5,A7,Th74,Th82;
        hence Sum R1 < Sum R2 by A4,A11,A10,A16,A17,XREAL_1:8;
      end;
    end;
    hence Sum R1 < Sum R2;
  end;
  then
A18: for i be Nat st P[i] holds P[i+1];
A19: P[0];
  for i be Nat holds P[i] from NAT_1:sch 2(A19,A18);
  hence thesis by A0;
end;
