
theorem Th37:
  for S,T being complete LATTICE for f being
directed-sups-preserving projection Function of T,T holds Image UPS(id S, f) =
  UPS(S, Image f)
proof
  let S,T be complete LATTICE;
  let f be directed-sups-preserving projection Function of T,T;
  reconsider If = Image f as complete LATTICE by YELLOW_2:35;
A1: (If)|^ the carrier of S is full SubRelStr of T |^ the carrier of S by
YELLOW16:39;
  UPS(S, If) is full SubRelStr of (Image f)|^ the carrier of S by Def4;
  then reconsider
  UPSIf=UPS(S, If) as full SubRelStr of T |^ the carrier of S by A1,WAYBEL15:1;
  UPS(S, T) is full SubRelStr of T |^ the carrier of S by Def4;
  then reconsider
  IUPS=Image UPS(id S,f) as full SubRelStr of T|^the carrier of S
  by WAYBEL15:1;
  the carrier of UPSIf = the carrier of IUPS
  proof
    thus the carrier of UPSIf c= the carrier of IUPS
    proof
      let x be object;
A2:   dom UPS(id S, f) = the carrier of UPS(S, T) by FUNCT_2:def 1;
      assume x in the carrier of UPSIf;
      then reconsider h=x as directed-sups-preserving Function of S, If by Def4
;
      the carrier of If c= the carrier of T by YELLOW_0:def 13;
      then
A3:   rng h c= the carrier of T;
      dom h = the carrier of S by FUNCT_2:def 1;
      then reconsider g=h as Function of S, T by A3,RELSET_1:4;
A4:   g is directed-sups-preserving
      proof
        let X be Subset of S;
        assume
A5:     X is non empty directed;
        thus g preserves_sup_of X
        proof
          reconsider hX = h.:X as non empty directed Subset of If by A5,
YELLOW_2:15;
          assume
A6:       ex_sup_of X,S;
          h preserves_sup_of X by A5,WAYBEL_0:def 37;
          then
A7:       sup (hX) = h.sup X by A6;
          thus
A8:       ex_sup_of g.:X,T by YELLOW_0:17;
          If is directed-sups-inheriting non empty full SubRelStr of T by
YELLOW_2:35;
          hence thesis by A7,A8,WAYBEL_0:7;
        end;
      end;
      then
A9:   g in the carrier of UPS(S, T) by Def4;
      UPS(id S, f).g = f*g*(id S) by A4,Def5
        .= h*(id S) by Th10
        .= g by FUNCT_2:17;
      then x in rng UPS(id S, f) by A9,A2,FUNCT_1:def 3;
      hence thesis by YELLOW_0:def 15;
    end;
    let x be object;
A10: rng f = the carrier of subrelstr rng f by YELLOW_0:def 15;
    assume x in the carrier of IUPS;
    then x in rng UPS(id S, f) by YELLOW_0:def 15;
    then consider a being object such that
A11: a in dom UPS(id S, f) and
A12: x = UPS(id S, f).a by FUNCT_1:def 3;
    reconsider a as directed-sups-preserving Function of S, T by A11,Def4;
A13: x = f*a*(id S) by A12,Def5
      .= f*a by FUNCT_2:17;
    then reconsider x0=x as directed-sups-preserving Function of S,T by
WAYBEL15:11;
A14: rng x0 c= the carrier of T;
    dom x0 = the carrier of S by FUNCT_2:def 1;
    then reconsider x1 = x0 as Function of S, If by A10,A13,A14,FUNCT_2:2
,RELAT_1:26;
    x1 is directed-sups-preserving
    proof
      let X be Subset of S;
      assume
A15:  X is non empty directed;
      thus x1 preserves_sup_of X
      proof
        reconsider hX = x0.:X as non empty directed Subset of T by A15,
YELLOW_2:15;
A16:    If is directed-sups-inheriting non empty full SubRelStr of T by
YELLOW_2:35;
        reconsider gX = x1.:X as non empty directed Subset of If by Th12,A15,
YELLOW_2:15;
        assume
A17:    ex_sup_of X,S;
        thus ex_sup_of x1.:X,If by YELLOW_0:17;
A18:    x0 preserves_sup_of X by A15,WAYBEL_0:def 37;
        then ex_sup_of x0.:X,T by A17;
        then sup (gX) = sup (hX) by A16,WAYBEL_0:7;
        hence thesis by A17,A18;
      end;
    end;
    hence thesis by Def4;
  end;
  hence thesis by YELLOW_0:57;
end;
