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 Th18:
  max(f,g) = (C,f)@ "\/" (C,g)@
proof
  set fg = (C,f)@ "\/" (C,g)@, R = RealPoset [. 0,1 .], J = C --> R;
A1: (RealPoset [. 0,1 .]) |^ C = product (C --> RealPoset [. 0,1 .]) by
YELLOW_1:def 5;
  now
    let c;
    ( for c being Element of C holds J.c is sup-Semilattice)& J.c =
    RealPoset [. 0 ,1 .];
    hence (@fg).c = ((C,f)@.c) "\/" ((C,g)@.c) by A1,Th12
      .= max(f.c, g.c);
  end;
  hence thesis by FUZZY_1:def 4;
end;
