
theorem
for A be Subset of REAL, F1,F2 be Interval_Covering of A, n,m be Nat
 st (for k be Nat st k <> n & k <> m holds F1.k = F2.k) &
    F1.n = F2.m & F1.m = F2.n holds
 for k be Nat st k >= n & k >= m holds (Ser (F1 vol)).k = (Ser (F2 vol)).k
proof
    let A be Subset of REAL, F1,F2 be Interval_Covering of A, n,m be Nat;
    assume that
A1:  for k be Nat st k <> n & k <> m holds F1.k = F2.k and
A2:  F1.n = F2.m and
A3:  F1.m = F2.n;

    let k be Nat;
    assume that
A4:  k >= n and
A5:  k >= m;
A6: n is Element of NAT & m is Element of NAT by ORDINAL1:def 12; then
    (F1 vol).n = diameter(F1.n)
  & (F1 vol).m = diameter(F1.m) by MEASURE7:def 4; then
A7: (F1 vol).n = (F2 vol).m
  & (F1 vol).m = (F2 vol).n by A2,A3,A6,MEASURE7:def 4;

    for k be Nat st k <> n & k <> m holds (F1 vol).k = (F2 vol).k
    proof
     let k be Nat;
A8:  k is Element of NAT by ORDINAL1:def 12;
     assume k <> n & k <> m; then
     F1.k = F2.k by A1; then
     (F1 vol).k = diameter (F2.k) by A8,MEASURE7:def 4;
     hence (F1 vol).k = (F2 vol).k by A8,MEASURE7:def 4;
    end;
    hence (Ser (F1 vol)).k = (Ser (F2 vol)).k
      by A4,A5,A7,Th50,MEASURE7:12;
end;
