
theorem Th52:
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 vol F1 = vol F2
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;

A4: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
A5:(F1 vol).n = (F2 vol).m & (F1 vol).m = (F2 vol).n
     by A2,A3,A4,MEASURE7:def 4;

A6:for k be Nat st k <> n & k <> m holds (F1 vol).k = (F2 vol).k
   proof
    let k be Nat;
A7: 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 A7,MEASURE7:def 4;
    hence (F1 vol).k = (F2 vol).k by A7,MEASURE7:def 4;
   end; then
A8:for k be Nat st k <> m & k <> n holds (F2 vol).k = (F1 vol).k;

   n >= m or m > n; then
   SUM(F1 vol) = SUM(F2 vol) by A5,A6,A8,Th51,MEASURE7:12; then
   vol F1 = SUM(F2 vol) by MEASURE7:def 6;
   hence vol F1 = vol F2 by MEASURE7:def 6;
end;
