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

theorem TMP6:
  for s1,s2 be Real_Sequence
    st s1 is nonnegative & s1,s2 are_fiberwise_equipotent
  holds s2 is nonnegative
proof
   let s1,s2 be Real_Sequence;
   assume that
A1: s1 is nonnegative and
A2: s1,s2 are_fiberwise_equipotent;
   consider H be Function such that
A3: dom H = dom s2 & rng H = dom s1 & H is one-to-one & s2=s1*H
     by A2,CLASSES1:77;
A4:dom H = NAT & rng H = NAT by A3,FUNCT_2:def 1;
   now let m be Nat;
A6: H.m in dom s1 by A3,A4,ORDINAL1:def 12,FUNCT_1:3;
    s2.m = s1.(H.m) by A3,A4,ORDINAL1:def 12,FUNCT_1:13;
    hence 0 <= s2.m by A1,A6;
   end;
   hence s2 is nonnegative;
end;
