reserve x,y,z for Real,
  R for real non empty RelStr,
  a,b for Element of R;
reserve C for non empty set,
  c for Element of C,
  f,g for Membership_Func of C,
  s,t for Element of FuzzyLattice C;

theorem Th16:
  (for c holds f.c <= g.c) iff (C,f)@ <<= (C,g)@
proof
A1: (RealPoset [. 0,1 .]) |^ C = product (C --> RealPoset [. 0,1 .]) by
YELLOW_1:def 5;
A2: (for c holds f.c <= g.c) implies (C,f)@ <<= (C,g)@
  proof
    set f9 = (C,f)@, g9 = (C,g)@;
    reconsider f9,g9 as Element of product (C --> RealPoset [. 0,1 .]) by
YELLOW_1:def 5;
    assume
A3: for c holds f.c <= g.c;
    for c holds f9.c <<= g9.c by A3,Th3;
    hence thesis by A1,WAYBEL_3:28;
  end;
  (C,f)@ <<= (C,g)@ implies for c holds f.c <= g.c
  proof
    reconsider ff = (C,f)@, gg = (C,g)@ as Element of product (C --> RealPoset
    [. 0,1 .]) by YELLOW_1:def 5;
    assume
A4: (C,f)@ <<= (C,g)@;
    let c;
    (C --> RealPoset [. 0,1 .]).c = RealPoset [. 0,1 .] & ff.c <<= gg.c by A1
,A4,WAYBEL_3:28;
    hence thesis by Th3;
  end;
  hence thesis by A2;
end;
