reserve n for Nat,
  a,b for Real,
  s for Real_Sequence;

theorem
  (for n holds s.n = a*n+b) implies for n holds Partial_Sums(s).n = (n+1
  )*(s.0 + s.n)/2
proof
  assume
A1: for n holds s.n = a*n+b;
  let n;
  Partial_Sums(s).n = a*(n+1)*n/2+n*b+b by A1,Lm14
    .= (n+1)*(n*a+b+b)/2
    .= (n+1)*(s.n+(a*0+b))/2 by A1
    .= (n+1)*(s.n+s.0)/2 by A1;
  hence thesis;
end;
