theorem Th7:
  i in Seg (Sum f) implies min(f,i)-'1 = min(f,i) - 1 & Sum(f| (min(
  f,i)-'1))<i
proof
  set F=min(f,i);
  assume
A1: i in Seg (Sum f);
  then F in dom f by Def1;
  then 1<=F by FINSEQ_3:25;
  hence
A2: F-'1=F-1 by XREAL_1:233;
  assume Sum(f| (F-'1))>=i;
  then F-'1>=F-0 by A1,Def1;
  hence thesis by A2,XREAL_1:10;
end;
