
theorem
  for S be non empty finite set, s be non empty FinSequence of S, n be Nat st
  n in dom s
  ex m be Nat st (freqSEQ(s)).m = frequency(s.n,s) & s.n=(canFS(S)).m
proof
  let S be non empty finite set, s be non empty FinSequence of S, n be Nat;
  set x=s.n;
  assume n in dom s;
  then x in rng s by FUNCT_1:3;
  then x in S;
  then x in rng (canFS S) by FUNCT_2:def 3;
  then consider m be object such that
A1: m in dom (canFS(S)) and
A2: x=(canFS(S)).m by FUNCT_1:def 3;
  reconsider m as Nat by A1;
  take m;
  Seg len canFS S = Seg card S by FINSEQ_1:93;
  then dom canFS S = Seg card S by FINSEQ_1:def 3;
  then
  ex xx be Element of S st (freqSEQ s).m =frequency(xx,s) & xx=(canFS S).
  m by A1,Th11;
  hence thesis by A2;
end;
