
theorem Th21:
  for L1,L2,L3 be non empty transitive antisymmetric RelStr for f
  be Function of L1,L2 st f is infs-preserving holds L2 is full infs-inheriting
SubRelStr of L3 & L3 is complete implies ex g be Function of L1,L3 st f = g & g
  is infs-preserving
proof
  let L1,L2,L3 be non empty transitive antisymmetric RelStr;
  let f be Function of L1,L2;
  assume that
A1: f is infs-preserving and
A2: L2 is full infs-inheriting SubRelStr of L3 and
A3: L3 is complete;
  the carrier of L2 c= the carrier of L3 by A2,YELLOW_0:def 13;
  then reconsider g = f as Function of L1,L3 by FUNCT_2:7;
  take g;
  thus f = g;
  now
    let X be Subset of L1;
    now
A4:   f preserves_inf_of X by A1,WAYBEL_0:def 32;
      assume
A5:   ex_inf_of X,L1;
      thus
A6:   ex_inf_of g.:X,L3 by A3,YELLOW_0:17;
      then "/\"(f.:X,L3) in the carrier of L2 by A2,YELLOW_0:def 18;
      hence inf (g.:X) = inf (f.:X) by A2,A6,YELLOW_0:63
        .= g.inf X by A5,A4,WAYBEL_0:def 30;
    end;
    hence g preserves_inf_of X by WAYBEL_0:def 30;
  end;
  hence thesis by WAYBEL_0:def 32;
end;
