reserve x,y,y1,y2 for set;
reserve C for non empty set;
reserve c for Element of C;
reserve f,h,g,h1 for Membership_Func of C;

theorem Th26:
  f\g = EMF(C) implies f c= g
proof
  assume
A1: f\g = EMF(C);
  let c;
A2: min(f.c,(1_minus g).c) = (EMF(C)).c by A1,FUZZY_1:5;
  per cases by A2,XXREAL_0:15;
  suppose
    f.c = (EMF(C)).c;
    hence thesis by FUZZY_1:16;
  end;
  suppose
A3: (1_minus g).c = (EMF(C)).c;
    g.c = (1_minus (1_minus g)).c .= 1 - (1_minus g).c by FUZZY_1:def 5
      .= (1_minus EMF(C)).c by A3,FUZZY_1:def 5
      .= (UMF(C)).c by FUZZY_1:40;
    hence thesis by FUZZY_1:16;
  end;
end;
