
theorem Th3:
  for S be non empty finite set,
  x be Element of S holds
  x in rng canFS(S) &
  ex n be Nat st n in dom (canFS S) & x=(canFS S).n & n in Seg (card S)
  proof
    let S be non empty finite set,
    x be Element of S;
    A1: x in S; then
    x in rng canFS(S) by FUNCT_2:def 3; then
    consider n be object such that
    A2: n in dom (canFS(S)) & x=(canFS(S)).n by FUNCT_1:def 3;
    reconsider n as Nat by A2;
    len canFS(S) = card (S) by FINSEQ_1:93; then
    n in Seg (card (S)) by A2,FINSEQ_1:def 3;
    hence thesis by A2,A1,FUNCT_2:def 3;
  end;
