
theorem Th8:
  for S being continuous complete non empty Poset, A being set st
A is_FreeGen_set_of S for h9 being CLHomomorphism of S, S st h9|A = id A holds
  h9 = id S
proof
  let S be continuous complete non empty Poset, A be set such that
A1: A is_FreeGen_set_of S;
  set L = S;
A2: A is Subset of L by A1,Th7;
  then
A3: (id L)|A = id A by FUNCT_3:1;
  dom id A = A & rng id A = A;
  then reconsider f = id A as Function of A,the carrier of L by A2,RELSET_1:4;
  consider h being CLHomomorphism of L, L such that
  h|A = f and
A4: for h9 being CLHomomorphism of L,L st h9|A = f holds h9 = h by A1;
A5: id L is CLHomomorphism of L, L by Th5;
  let h9 be CLHomomorphism of S, S;
  assume h9|A = id A;
  hence h9 = h by A4
    .= id L by A4,A5,A3;
end;
