reserve n,m,k for Nat,
  x,y for set,
  r for Real;
reserve C,D for non empty finite set,
  a for FinSequence of bool D;

theorem Th5:
  for a be ascending terms've_same_card_as_number FinSequence of
  bool D holds for n,m holds n in dom a & m in dom a & n<>m implies a.n <> a.m
proof
  let A be ascending terms've_same_card_as_number FinSequence of bool D;
  let n,m;
  assume that
A1: n in dom A and
A2: m in dom A and
A3: n <> m & A.n = A.m;
A4: 1 <= m & m <= len A by A2,FINSEQ_3:25;
A5: 1 <= n & n <= len A by A1,FINSEQ_3:25;
  reconsider Am = A.m as finite set by A4,Lm2;
  card(Am)=m by A4,Def1;
  hence contradiction by A3,A5,Def1;
end;
