reserve k, l, m, n, i, j for Nat,
  K, N for non empty Subset of NAT,
  Ke, Ne, Me for Subset of NAT,
  X,Y for set;
reserve f for Function of Segm n,Segm k;
reserve x,y for set;

theorem
  for F be Function of Ne,Ke st rng F is finite for I1,I2 be Function of
Ne,Ke, P1,P2 be Permutation of rng F st F=P1*I1 & F=P2*I2 & rng F=rng I1 & rng
  F=rng I2 & I1 is 'increasing & I2 is 'increasing holds P1=P2 & I1=I2
proof
  let F be Function of Ne,Ke such that
A1: rng F is finite;
  let I1,I2 be Function of Ne,Ke, P1,P2 be Permutation of rng F such that
A2: F=P1*I1 and
A3: F=P2*I2 and
A4: rng F=rng I1 and
A5: rng F=rng I2 and
A6: I1 is 'increasing and
A7: I2 is 'increasing;
A8: rng F ={} implies rng F={};
  then
A9: dom P2=rng F by FUNCT_2:def 1;
  dom P1=rng F by A8,FUNCT_2:def 1;
  hence thesis by A1,A2,A3,A4,A5,A6,A7,A9,Th64;
end;
