
theorem Th11:
  for S be non empty finite set, s be non empty FinSequence of S, n be Nat
  st n in Seg (card S)
  ex x be Element of S st (freqSEQ(s)).n=frequency(x,s)& x=(canFS(S)).n
proof
  let S be non empty finite set, s be non empty FinSequence of S, n be Nat;
  set y =(len s)*(FDprobSEQ(s)).n;
A1: rng canFS S = S by FUNCT_2:def 3;
  assume
A2: n in Seg card S;
  then
A3: n in dom freqSEQ(s) by Def9;
  Seg len canFS S = Seg card S by FINSEQ_1:93;
  then n in dom canFS S by A2,FINSEQ_1:def 3;
  then (canFS S).n is Element of S by A1,FUNCT_1:3;
  then consider a be Element of S such that
A4: a=(canFS S).n;
  take a;
  n in dom FDprobSEQ(s) by A2,Def3;
  then y=(len s)*FDprobability(a,s) by A4,Def3;
  then y= frequency(a,s) by XCMPLX_1:87;
  hence thesis by A3,A4,Def9;
end;
