reserve x,y for object,
        D,D1,D2 for non empty set,
        i,j,k,m,n for Nat,
        f,g for FinSequence of D*,
        f1 for FinSequence of D1*,
        f2 for FinSequence of D2*;
reserve f for complex-valued Function,
        g,h for complex-valued FinSequence;

theorem
  for f1,f2 be one-to-one natural-valued FinSequence st
    n > 1 & (n|^f1).1 + (n|^f1,2)+... = (n|^f2).1 + (n|^f2,2)+...
  holds rng f1 = rng f2
proof
  let f1,f2 be one-to-one natural-valued FinSequence such that
  A1:   n > 1 & (n|^f1).1+(n|^f1,2)+... = (n|^f2).1+(n|^f2,2)+...;
  A2:rng f1 c= NAT & rng f2 c= NAT by VALUED_0:def 6;
  then reconsider F1=f1,F2=f2 as FinSequence of REAL by FINSEQ_1:def 4;
  set s1=sort_a F1,s2=sort_a F2;
  A3:F1,s1 are_fiberwise_equipotent & F2,s2 are_fiberwise_equipotent
    by RFINSEQ2:def 6;
  A4:rng s1=rng f1 by RFINSEQ2: def 6,CLASSES1:75;
  then A5:s1 is natural-valued by A2,VALUED_0:def 6;
  n|^F1,n|^s1 are_fiberwise_equipotent by A3,A1,A5,Th29;
  then A8: Sum (n|^F1) = Sum(n|^s1) by RFINSEQ:9;
  A9:rng s2=rng f2 by RFINSEQ2:def 6,CLASSES1:75;
  then A10:s2 is natural-valued by A2,VALUED_0:def 6;
  A12:s2 is natural-valued by A9,A2,VALUED_0:def 6;
  n|^F2,n|^s2 are_fiberwise_equipotent by A10,A3,A1,Th29;
  then A14: Sum (n|^F2) = Sum(n|^s2) by RFINSEQ:9;
  A15:for e1,e2 be ExtReal st e1 in dom s1 & e2 in dom s1 & e1 < e2 holds
    s1.e1 < s1.e2
    proof
    let e1,e2 be ExtReal;
    assume A16:e1 in dom s1 & e2 in dom s1 & e1 < e2;
    then A17:s1.e1 <=s1.e2 by INTEGRA2:2;
    assume A18: s1.e1 >= s1.e2;
    consider H be Function such that
    A19:dom H = dom s1 & rng H = dom F1 & H is one-to-one & s1 = F1 *H
      by A3,CLASSES1:77;
    s1 is one-to-one by A19;
    hence thesis by A18,A17,XXREAL_0:1,A16;
  end;
  for e1,e2 be ExtReal st e1 in dom s2 & e2 in dom s2 & e1 < e2 holds
    s2.e1 < s2.e2
  proof
    let e1,e2 be ExtReal;
    assume A20:e1 in dom s2 & e2 in dom s2 & e1 < e2;
    then A21:s2.e1 <=s2.e2 by INTEGRA2:2;
    assume A22: s2.e1 >= s2.e2;
    consider H be Function such that
    A23:dom H = dom s2 & rng H = dom F2 & H is one-to-one & s2 = F2 *H
      by A3,CLASSES1:77;
    s2 is one-to-one by A23;
    hence thesis by A22,A21,XXREAL_0:1,A20;
  end;
  then A24:s2 is increasing by VALUED_0:def 13;
  A25:Sum (n|^s1) = (n|^s1).1+(n|^s1,2)+... by Th22;
  Sum (n|^f1) = (n|^f1).1+(n|^f1,2)+... by Th22;
  then Sum (n|^s1) = Sum (n|^s2) by Th22,A1,A8,A14;
  then
    (n|^s1).1+(n|^s1,2)+... = (n|^s2).1+(n|^s2,2)+... &
    s1 is increasing natural-valued
    by A15,VALUED_0:def 13,A25,Th22,A4,A2,VALUED_0:def 6;
  hence thesis by A1,A12,A24,Th27,A9,A4;
end;
