
theorem Th12:
  for S be finite set, s be FinSequence of S holds
  freqSEQ (s) =(len s)* ((FDprobSEQ(s)))
proof
  let S be finite set, s be FinSequence of S;
A1: dom ((len s) (#) ( (FDprobSEQ(s)))) =dom (FDprobSEQ(s)) by VALUED_1:def 5
    .=Seg (card (S)) by Def3
    .= dom (freqSEQ (s)) by Def9;
  now
    let m be Nat;
    assume
A2: m in dom ((len s) (#) ( (FDprobSEQ(s))));
    hence ((len s) (#) ( (FDprobSEQ(s)))).m =(len s)* ( (FDprobSEQ(s))).m by
VALUED_1:def 5
      .= (freqSEQ (s)).m by A1,A2,Def9;
  end;
  hence freqSEQ s = (len s) (#) (FDprobSEQ s) by A1;
end;
