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 Th27:
  min(f,g) = EMF(C) implies f\g = f
proof
A1: C = dom f by FUNCT_2:def 1;
  assume
A2: min(f,g) = EMF(C);
A3: for x being Element of C st x in C holds min(f,1_minus g).x = f.x
  proof
    let x be Element of C;
A4: (EMF C).x = min(f.x,g.x) by A2,FUZZY_1:5;
    per cases by A4,XXREAL_0:15;
    suppose
A5:   f.x = (EMF(C)).x;
      min(f,1_minus g).x = min(f.x,(1_minus g).x) by FUZZY_1:5
        .= min(1_minus g,EMF(C)).x by A5,FUZZY_1:5
        .= (EMF(C)).x by FUZZY_1:18;
      hence thesis by A5;
    end;
    suppose
A6:   g.x = (EMF(C)).x;
      min(f,1_minus g).x = min(f.x,(1_minus g).x) by FUZZY_1:5
        .= min(f.x,1 - (EMF(C)).x) by A6,FUZZY_1:def 5
        .= min(f.x,(1_minus EMF(C)).x) by FUZZY_1:def 5
        .= min(f.x,(UMF(C)).x) by FUZZY_1:40
        .= min(f,UMF(C)).x by FUZZY_1:5;
      hence thesis by FUZZY_1:18;
    end;
  end;
  C = dom min(f,1_minus g) by FUNCT_2:def 1;
  hence thesis by A1,A3,PARTFUN1:5;
end;
