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;

theorem
  f = id n & n > 0 implies f is "increasing
proof
  assume that
A1: f = id n and
A2: n>0;
A3: ex x be object st x in n by A2,XBOOLE_0:def 1;
A4: now
    let l,m such that
A5: l in rng f and
A6: m in rng f and
A7: l<m;
A8: ex x be object st f"{l} = {x} by A1,A5,FUNCT_1:74;
A9: l in {l} by TARSKI:def 1;
    consider l9 be object such that
A10: l9 in dom f and
A11: f.l9=l by A5,FUNCT_1:def 3;
    l = l9 by A1,A10,A11,FUNCT_1:18;
    then l in f"{l} by A10,A11,A9,FUNCT_1:def 7;
    then f"{l}={l} & l in NAT by A8,TARSKI:def 1;
    then
A12: min* f"{l} = l by Th5;
A13: m in {m} by TARSKI:def 1;
A14: ex x be object st f"{m} = {x} by A1,A6,FUNCT_1:74;
    consider m9 be object such that
A15: m9 in dom f and
A16: f.m9=m by A6,FUNCT_1:def 3;
    m = m9 by A1,A15,A16,FUNCT_1:18;
    then m in f"{m} by A15,A16,A13,FUNCT_1:def 7;
    then f"{m}={m} & m in NAT by A14,TARSKI:def 1;
    hence min* f"{l} < min* f"{m} by A7,A12,Th5;
  end;
  rng f=n by A1;
  hence thesis by A3,A4;
end;
