 reserve Rseq, Rseq1, Rseq2 for Function of [:NAT,NAT:],REAL;

theorem
  for s1,s2 being Real_Sequence st
    s1,s2 are_fiberwise_equipotent & s1 is nonnegative & s1 is summable
  holds s2 is summable & Sum s1 = Sum s2
proof
   let s1,s2 be Real_Sequence;
   assume that
A1: s1,s2 are_fiberwise_equipotent & s1 is nonnegative and
A2: s1 is summable;
   Partial_Sums s1 is bounded_above by A2,A1,SERIES_1:17; then
   consider r be Real such that
A5: for n being Nat holds (Partial_Sums s1).n < r by SEQ_2:def 3;
A3:for n being Nat holds 0<=s2.n by A1,TMP6,RINFSUP1:def 3;
B3:now let n be Nat;
    consider m be Nat such that
A6:  (Partial_Sums s2).n <= (Partial_Sums s1).m by A1,TMP6,SH8;
    thus (Partial_Sums s2).n < r by A5,A6,XXREAL_0:2;
   end; then
B2:Partial_Sums s2 is bounded_above by SEQ_2:def 3;
   hence A7: s2 is summable by A3,SERIES_1:17;
A8:Partial_Sums s1 is bounded_above
 & Partial_Sums s2 is bounded_above by A2,B3,SEQ_2:def 3,A1,SERIES_1:17;
   now let n be Nat;
    consider m be Nat such that
A11: (Partial_Sums s1).n <= (Partial_Sums s2).m by A1,SH8;
    (Partial_Sums s2).m <= lim Partial_Sums s2
      by A3,B2,SERIES_1:16,SEQ_4:37;
    hence (Partial_Sums s1).n <= lim Partial_Sums s2 by A11,XXREAL_0:2;
   end; then
   lim Partial_Sums s1 <= lim Partial_Sums s2
     by A2,SERIES_1:def 2,PREPOWER:2; then
   Sum s1 <= lim Partial_Sums s2 by SERIES_1:def 3; then
A12:Sum s1 <= Sum s2 by SERIES_1:def 3;
   now let m be Nat;
    consider n be Nat such that
A13: (Partial_Sums s2).m <= (Partial_Sums s1).n by A1,TMP6,SH8;
    (Partial_Sums s1).n <= lim Partial_Sums s1
      by A8,A1,SERIES_1:16,SEQ_4:37;
    hence (Partial_Sums s2).m <= lim Partial_Sums s1 by A13,XXREAL_0:2;
   end; then
   lim Partial_Sums s2 <= lim Partial_Sums s1
     by A7,SERIES_1:def 2,PREPOWER:2; then
   Sum s2 <= lim Partial_Sums s1 by SERIES_1:def 3; then
   Sum s2 <= Sum s1 by SERIES_1:def 3;
   hence Sum s1 = Sum s2 by A12,XXREAL_0:1;
end;
